5-manifolds: 1-connected

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1 Introduction

Let

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${{Authors|Diarmuid Crowley}} ==Introduction== <wikitex refresh>; Let $\mathcal{M}_{5}$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]], [[wikipedia:Differentiable_manifold|smooth]], [[wikipedia:Simply-connected|simply-connected]] [[wikipedia:5-manifold|5-manifolds]] $M$ and let $\mathcal{M}_5^{\text{Spin}}\subset \mathcal{M}_5$ be the subset of diffeomorphism classes of [[wikipedia:Spin_manifold|spinable manifolds]]. The calculation of $\mathcal{M}_5^\Spin$ was first obtained by Smale {{cite|Smale1962}} and was one of the first applications of the [[wikipedia:H-cobordism|h-cobordism theorem]]. A little latter Barden {{cite|Barden1965}} devised an elegant surgery argument and applied results of {{cite|Wall1964}} on the diffeomorphism groups of 4-manifolds to give an explicit and complete classification of all of $\mathcal{M}_{5}$. Simply-connected 5-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected 5-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that not every simply-connected 5-dimensional [[wikipedia:Poincaré_space|Poincaré space]] is smoothable. The classification of simply-connected 5-dimensional Poincaré spaces was achieved by Stöcker {{cite|Stöcker1982}}. </wikitex> == Constructions and examples == <wikitex>; We first list some familiar 5-manifolds using Barden's notation: * $X_0 \coloneq S^5$. * $M_\infty \coloneq S^2 \times S^3$. * $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$. * $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standard inclusion of $\SO_3 \rightarrow SU_3$. In {{cite|Barden1965|Section 1}} a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles $M \cong W \cup_f W$ where $W$ is a certain simply connected $-manifold with boundary $\partial W$ a simply-connected $-manifold and $f: \partial W \cong \partial W$ is a diffeomorphism. Barden used results of {{cite|Wall1964}} to show that diffeomorphisms realising the required ismorphisms of $H_2(\partial W)$ exist. <!--* !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties. --> </wikitex> === The general spin case === <wikitex>; Next we present a construction of simply-connected spin 5-manifolds. ''A priori'' the construction depends upon choices, but applying Theorem \ref{thm:Barden-realise} below shows that the choices do not affect the diffeomorphism type of the manifold constructed. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex with only $-cells and $-cells and so there is an embedding $X_G\to\Rr^6$. Let $N_G$ be a [[regularneighbourhood|regular neighbourhood]] of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N_G$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$ where $\sharp_r$ denotes the $r$-fold [[Connected sum|connected sum]]. </wikitex> === The general non-spin case === <wikitex>; For the non-spin case we construct only those manifolds which are boundaries of $-manifolds. As in the spin case, the construction depends ''a priori'' on choices, but Theorem \ref{thm:Barden-realise} entails that these choices do not affect the diffeomorphism type of the manifold constructed. Let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and second [[#Invariants|Stiefel-Whitney class]] $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-spin manifold as described above. </wikitex> == Invariants == <wikitex>; Consider the following invariants of a closed simply-connected $-manifold $M$: * $H_2(M)$ be the second integral homology group of $M$, * $w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second [[wikipedia:Stiefel–Whitney_class|Stiefel-Whitney class]] of $M$, $w_2 \in H^2(M; \Zz_2)$, * $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that ^r \cdot x = 0$ for some $x \in w_2^{-1}(1)$. If $M$ is spinable we set $h(M) = 0$. For example, the manifold $X_{\infty}$ has invariants $H_2(X_{\infty}) \cong \Z$, non-trivial $w_2$ and $h(X_{\infty}) = \infty$. The Wu-manifold, $X_{-1}$, has invariants $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$. The above list is the minimal list of invariants required to give the classification of closed simply-connected $-manifolds: see Theorem \ref{thm:minimal_classification} below. In addition we mention two further invariants of $M$: * $w_3 \in H^3(M; \Zz_2)$, the third Stiefel-Whitney class, * $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the [[Linking form|linking form]] of $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$, the torsion subgroup of $H_2(M)$. By \cite{Milnor&Stasheff1974|Problem 8-A}, $w_3 = Sq^1(w_2)$, and so $w_2$ determines $w_3$. By {{cite|Wall1962|Proposition 1 & 2}}, the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$. The classification of anti-symmetric linking forms is rather simple and this leads to the fact that one only needs to list the extended natural number $h(M)$ in order to obtain a complete list of invariants of simply-connected $-manifolds: This point is clarified in the following sub-section where we report on the classification of anti-symmetric linking forms. </wikitex> === Linking forms === <wikitex>; An abstract non-singular anti-symmetric linking form on a finite abelian group $H$ is a bi-linear function $$ b \co H \times H \rightarrow \Qq/\Zz $$ such that $b(x, y) = 0$ for all $y \in H$ if and only if $x = 0$ and $b(x, y) = -b(y, x)$ for all pairs $x$ and $y$. For example, if $C_k$ denotes the cyclic group of order $k$, we have the following linking forms specified by their linking matricies, $$ b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right), $$ $$ b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \-\frac{1}{k} & ~0 \end{array} \right),$$ $$ b_j(C_{2^j}) : (C_{2^j} \oplus C_{2^j}) \times (C_{2^j} \oplus C_{2^j}) \to \Qq/\Zz, \quad \left( \begin{array}{cc} \frac{1}{2} & ~\frac{1}{2^j} \ -\frac{1}{2^j} & ~0 \end{array} \right).$$ If $T = \oplus_{i=1}^r C_{k_r}$ is the sum of cyclic groups we shall write $b_0(T)$ for the sum $\oplus_{i=1}^r b_0(C_{k_r})$. By {{cite|Wall1963|Theorem 3}}, all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above and indeed such linking forms are classified up to isomorphism by the homomorphism $$ w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$ Moreover $H$ must be isomorphic to $T \oplus T$ or $T \oplus T \oplus \Zz_2$ for some finite group $T$ with $b(x,x) = 1/2$ if $x$ generates the $\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold $M$ determines the isomorphism class of the linking form $b_M$ and we see that the torsion subgroup of $H_2(M)$ is of the form $TH_2(M) \cong T \oplus T$ if $h(M) \neq 1$ or $TH_2(M) \cong T \oplus T \oplus \Zz_2$ if $h(M) = 1$ in which case the $\Zz_2$ summand is an orthogonal summand of $b_M$. </wikitex> === Values for constructions === <wikitex>; The spin manifolds $M_G$ all have vanishing $w_2$ of course and so by Wall's classification of linking forms we see that the linking form of $M_G$ is the linking form $b_0(TG)$. As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$ have $w_2$ given by projecting to $G$ and then applying $\omega$: $$ w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$$ If $M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$. </wikitex> == Classification == <wikitex>; We first present the most economical classifications of $\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$. {{beginthm|Theorem|{{cite|Smale1962|Theorem p.38}}}} There is a bijective correspondence $$\mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$ {{endthm}} {{beginthm|Theorem|{{cite|Barden1965}}}} \label{thm:minimal_classification} The mapping $$\mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$ is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$. {{endthm}} The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms. {{beginthm|Theorem|{{cite|Barden1965|Theorem 2.2}}}}\label{thm:Barden-realise} Let $M_0$ and $M_1$ be simply-connected, closed, smooth 5-manifolds and let $A\co H_2(M_0) \cong H_2(M_1)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $A$ is realised by a diffeomorphism $f_A : M_0 \cong M_1$. {{endthm}} This theorem can re-phrased in categorical language as follows. * Let $\mathcal{Q}_5$ be the groupoid with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is an anti-symmetric non-singular linking form and $w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$ for all $x \in TG$. The morphisms of $\mathcal{Q}_5$ are isomorphisms of abelian groups commuting with both $w$ and $b$. * Let $\widetilde{\mathcal{M}}_5$ be the groupoid with objects simply-connected closed smooth $-manifolds embedded in some fixed $\Rr^N$ for $N$ large and morphisms isotopy classes of diffeomorphisms. <!-- * Let $(b, w_2) \co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$ be the funtor which maps $M$ to $(H_2(M), b_M, w_2(M))$ and $f \co M_0 \cong M_1$ to the induced map on $H_2$.--> * Consider the functor $$(b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$$ {{beginthm|Theorem|{{cite|Barden1965}}}} The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$ is a detecting functor. That is, it induces a bijection on isomorphism classes of objects. {{endthm}} </wikitex> === Enumeration === <wikitex>; We first give Barden's enumeration of the set $\mathcal{M}^{}_5$, {{cite|Barden1965|Theorem 2.3}}. * $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$. * For \mathcal{M}_{5}$ be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds

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$M$ and let

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$\mathcal{M}_5^{\text{Spin}}\subset \mathcal{M}_5$ be the subset of diffeomorphism classes of spinable manifolds. The calculation of

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$\mathcal{M}_5^\Spin$ was first obtained by Smale [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden [Barden1965] devised an elegant surgery argument and applied results of [Wall1964] on the diffeomorphism groups of 4-manifolds to give an explicit and complete classification of all of

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$\mathcal{M}_{5}$.

Simply-connected 5-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected 5-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that not every simply-connected 5-dimensional Poincaré space is smoothable. The classification of simply-connected 5-dimensional Poincaré spaces was achieved by Stöcker [Stöcker1982].

2 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$.
• $M_\infty \coloneq S^2 \times S^3$$M_\infty \coloneq S^2 \times S^3$.
• $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, the total space of the non-trivial $S^3$$S^3$-bundle over $S^2$$S^2$.
• $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standard inclusion of $\SO_3 \rightarrow SU_3$$\SO_3 \rightarrow SU_3$.

In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles $M \cong W \cup_f W$$M \cong W \cup_f W$ where $W$$W$ is a certain simply connected $5$$5$-manifold with boundary $\partial W$$\partial W$ a simply-connected $4$$4$-manifold and $f: \partial W \cong \partial W$$f: \partial W \cong \partial W$ is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of $H_2(\partial W)$$H_2(\partial W)$ exist.

2.1 The general spin case

Next we present a construction of simply-connected spin 5-manifolds. A priori the construction depends upon choices, but applying Theorem 4.3 below shows that the choices do not affect the diffeomorphism type of the manifold constructed. Note that all homology groups are with integer coefficients.

Given a finitely generated abelian group $G$$G$, let $X_G$$X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$$H_2(X_G) = G$. The space $X_G$$X_G$ may be realised as a finite CW-complex with only $2$$2$-cells and $3$$3$-cells and so there is an embedding $X_G\to\Rr^6$$X_G\to\Rr^6$. Let

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$N_G$ be a regular neighbourhood of $X_G\subset\Rr^6$$X_G\subset\Rr^6$ and let

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$M_G$ be the boundary of

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$N_G$. Then

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$M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$$H_2(M_G)\cong G \oplus TG$ where $TG$$TG$ is the torsion subgroup of $G$$G$. For example, $M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$$M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$ where $\sharp_r$$\sharp_r$ denotes the $r$$r$-fold connected sum.

2.2 The general non-spin case

For the non-spin case we construct only those manifolds which are boundaries of $6$$6$-manifolds. As in the spin case, the construction depends a priori on choices, but Theorem 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.

Let $(G, w)$$(G, w)$ be a pair with $w\co G \to\Zz_2$$w\co G \to\Zz_2$ a surjective homomorphism and $G$$G$ as above. We shall construct a non-spin 5-manifold $M_{(G, w)}$$M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$$H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$$w_2$ given by $w$$w$ composed with the projection $G \oplus TG \to G$$G \oplus TG \to G$.

If $(G, w) = (\Zz, 1)$$(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$$N_{(\Zz, 1)}$ be the non-trivial $D^4$$D^4$-bundle over $S^2$$S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$$\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$$(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$$N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$$N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$$M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$.

In the general case, present $G = F/i(R)$$G = F/i(R)$ where $i \co R \to F$$i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$$(G, w)$ to $(F, \bar w)$$(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$$F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$$\{r_1, \dots, r_n \}$ is a basis for $R$$R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$$i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$$N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$$N_{(F, \bar w)}$ along spheres representing $i(r_i)$$i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$$M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$$M_{(G, w)}$ is a non-spin manifold as described above.

3 Invariants

Consider the following invariants of a closed simply-connected $5$$5$-manifold

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$M$:

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  $H_2(M)$ be the second integral homology group of

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  $M$,
• $w_2 \co H_2(M) \rightarrow \Zz_2$$w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$$w_2 \in H^2(M; \Zz_2)$,
• $h(M) \in \Nn \cup \{\infty\}$$h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$$r$ such that $2^r \cdot x = 0$$2^r \cdot x = 0$ for some $x \in w_2^{-1}(1)$$x \in w_2^{-1}(1)$. If

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  $M$ is spinable we set $h(M) = 0$$h(M) = 0$.

For example, the manifold $X_{\infty}$$X_{\infty}$ has invariants $H_2(X_{\infty}) \cong \Z$$H_2(X_{\infty}) \cong \Z$, non-trivial $w_2$$w_2$ and $h(X_{\infty}) = \infty$$h(X_{\infty}) = \infty$. The Wu-manifold, $X_{-1}$$X_{-1}$, has invariants $H_2(X_{-1}) = \Zz_2$$H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$$w_2$ and $h(X_{-1}) = 1$$h(X_{-1}) = 1$.

The above list is the minimal list of invariants required to give the classification of closed simply-connected $5$$5$-manifolds: see Theorem 4.2 below.

In addition we mention two further invariants of

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$M$:

• $w_3 \in H^3(M; \Zz_2)$$w_3 \in H^3(M; \Zz_2)$, the third Stiefel-Whitney class,
• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$$b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$$TH_2(M)$, the torsion subgroup of

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  $H_2(M)$.

By [Milnor&Stasheff1974, Problem 8-A], $w_3 = Sq^1(w_2)$$w_3 = Sq^1(w_2)$, and so $w_2$$w_2$ determines $w_3$$w_3$.

By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity $b_M(x, x) = w_2(x)$$b_M(x, x) = w_2(x)$ where we regard $w_2(x)$$w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$$\{0, 1/2\} \subset \Qq/\Zz$. The classification of anti-symmetric linking forms is rather simple and this leads to the fact that one only needs to list the extended natural number $h(M)$$h(M)$ in order to obtain a complete list of invariants of simply-connected $5$$5$-manifolds: This point is clarified in the following sub-section where we report on the classification of anti-symmetric linking forms.

3.1 Linking forms

An abstract non-singular anti-symmetric linking form on a finite abelian group $H$$H$ is a bi-linear function

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$$\displaystyle b \co H \times H \rightarrow \Qq/\Zz$$

such that

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$b(x, y) = 0$ for all

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$y \in H$ if and only if

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$x = 0$ and

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$b(x, y) = -b(y, x)$ for all pairs $x$$x$ and $y$$y$. For example, if $C_k$$C_k$ denotes the cyclic group of order $k$$k$, we have the following linking forms specified by their linking matricies,

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$$\displaystyle b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right),$$

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$$\displaystyle b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),$$

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$$\displaystyle b_j(C_{2^j}) : (C_{2^j} \oplus C_{2^j}) \times (C_{2^j} \oplus C_{2^j}) \to \Qq/\Zz, \quad \left( \begin{array}{cc} \frac{1}{2} & ~\frac{1}{2^j} \\ -\frac{1}{2^j} & ~0 \end{array} \right).$$

If

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$T = \oplus_{i=1}^r C_{k_r}$ is the sum of cyclic groups we shall write

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$b_0(T)$ for the sum

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$\oplus_{i=1}^r b_0(C_{k_r})$.

By [Wall1963, Theorem 3], all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above and indeed such linking forms are classified up to isomorphism by the homomorphism

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$$\displaystyle w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$

Moreover $H$$H$ must be isomorphic to

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$T \oplus T$ or

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$T \oplus T \oplus \Zz_2$ for some finite group $T$$T$ with

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$b(x,x) = 1/2$ if $x$$x$ generates the $\Zz_2$$\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold

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$M$ determines the isomorphism class of the linking form

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$b_M$ and we see that the torsion subgroup of

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$H_2(M)$ is of the form

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$TH_2(M) \cong T \oplus T$ if

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$h(M) \neq 1$ or

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$TH_2(M) \cong T \oplus T \oplus \Zz_2$ if

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$h(M) = 1$ in which case the $\Zz_2$$\Zz_2$ summand is an orthogonal summand of

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$b_M$.

3.2 Values for constructions

The spin manifolds

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$M_G$ all have vanishing $w_2$$w_2$ of course and so by Wall's classification of linking forms we see that the linking form of

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$M_G$ is the linking form $b_0(TG)$$b_0(TG)$.

As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$$M_{(G, \omega)}$ have $w_2$$w_2$ given by projecting to $G$$G$ and then applying $\omega$$\omega$:

$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$ $$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$$

If $M_{(G, \omega)}$$M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$$h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$$b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$$TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$$M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$$b_{M_{(G, \omega)}} = b_0(TG)$.

4 Classification

We first present the most economical classifications of $\mathcal{M}^\Spin_5$$\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$$\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$$G$ with torsion subgroup $TG \cong H \oplus H \oplus C$$TG \cong H \oplus H \oplus C$ where $C$$C$ is trivial or $C \cong \Zz_2$$C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$.

Theorem 4.1 [Smale1962, Theorem p.38]. There is a bijective correspondence

$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$ $$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 4.2 [Barden1965]. The mapping

$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$ $$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$$([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$$[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$$n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

Theorem 4.3 [Barden1965, Theorem 2.2]. Let $M_0$$M_0$ and $M_1$$M_1$ be simply-connected, closed, smooth 5-manifolds and let $A\co H_2(M_0) \cong H_2(M_1)$$A\co H_2(M_0) \cong H_2(M_1)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $A$$A$ is realised by a diffeomorphism $f_A : M_0 \cong M_1$$f_A : M_0 \cong M_1$.

This theorem can re-phrased in categorical language as follows.

• Let $\mathcal{Q}_5$$\mathcal{Q}_5$ be the groupoid with objects $(G, b, w)$$(G, b, w)$ where $G$$G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$$b \co TG \times TG \to \Qq/\Zz$ is an anti-symmetric non-singular linking form and $w\co G \to \Zz_2$$w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$$w(x) = b(x, x)$ for all $x \in TG$$x \in TG$. The morphisms of $\mathcal{Q}_5$$\mathcal{Q}_5$ are isomorphisms of abelian groups commuting with both $w$$w$ and $b$$b$.

• Let $\widetilde{\mathcal{M}}_5$$\widetilde{\mathcal{M}}_5$ be the groupoid with objects simply-connected closed smooth $5$$5$-manifolds embedded in some fixed $\Rr^N$$\Rr^N$ for $N$$N$ large and morphisms isotopy classes of diffeomorphisms.
• Consider the functor

$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$ $$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$$

Theorem 4.4 [Barden1965]. The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$$(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$ is a detecting functor. That is, it induces a bijection on isomorphism classes of objects.

4.1 Enumeration

We first give Barden's enumeration of the set $\mathcal{M}^{}_5$$\mathcal{M}^{}_5$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$$M_\infty\coloneq S^2 \times S^3$, $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$$1 < k < \infty$, $M_k = M_{\Zz_k}$$M_k = M_{\Zz_k}$ is the spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$$H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$$1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$$X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$$H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$ $$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$$-1 \leq j \leq \infty$, $1 < k_i$$1 < k_i$, $k_i$$k_i$ divides $k_{i+1}$$k_{i+1}$ or $k_{i+1} = \infty$$k_{i+1} = \infty$ and $\sharp$$\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$$X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$$X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$$(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5$$\mathcal{M}_5$ as a disjoint union

$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$ $$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$$

where the last two sets denote the diffeomorphism classes of non-spinable 5-manifolds which are respectively boundaries and not boundaries. Then

$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$ $$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$$

5 Further discussion

• As the invariants which classify simply-connected closed oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
• By the construction above every simply-connected closed smooth spinable $5$$5$-manifold embeds into $\Rr^6$$\Rr^6$.

• As the invariants for $-M$$-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

• Barden's results have been nicely discussed and re-proven by Zhubr [Zhubr2001].

5.1 Bordism groups

As $\mathcal{M}_5^\Spin = \{[M_G]\}$$\mathcal{M}_5^\Spin = \{[M_G]\}$,

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$M_G = \partial N_G$ and

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$M_G$ admits a unique spin structure which extends to

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$N_G$ we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group $\Omega_5^\Spin$$\Omega_5^\Spin$ vanishes. The bordism group

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$\Omega_5^{\SO}$ is isomorphic to $\Zz_2$$\Zz_2$, see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number

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$\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

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$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,$$

and

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$w_2(X_{-1}) \neq 0$. It follows that

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$w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and so we have that

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$\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that

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$[X_{-1}]$ is the generator of

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$\Omega_5^{\SO}$ and that a closed, smooth simply-connected 5-manifold

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$M$ is not a boundary if and only if it is diffeomorphic to

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$X_{-1} \sharp M_0$ where $M_0$$M_0$ is a Spin manifold.

5.2 Curvature and contact structures

Every manifold $\sharp_r(S^2 \times S^3)$$\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature by [Boyer&Galicki2006].

The following theorem is an immediate consequence of [Geiges1991, Theorem 1 & Lemma 7].

Theorem 5.1 [Geiges1991].

A simply connected $5$$5$-manifold

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$M$ admits a contact structure if and only if $w_2(M) \in H^2(M; \Zz_2)$$w_2(M) \in H^2(M; \Zz_2)$ has an integral lift in $H^2(M; \Zz)$$H^2(M; \Zz)$. Hence

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$M$ admits a contact structure if and only if $h(M) = 0$$h(M) = 0$ or $\infty$$\infty$; equivalently

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$M$ admits a contact structure if and only if $M \in \mathcal{M}_5^\Spin$$M \in \mathcal{M}_5^\Spin$ or $M \cong M_0 \sharp X_{\infty}$$M \cong M_0 \sharp X_{\infty}$ where $M_0 \in \mathcal{M}_5^\Spin$$M_0 \in \mathcal{M}_5^\Spin$.

Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of $TH_2(M)$$TH_2(M)$ prime to 3 was proven in [Thomas1986].

5.3 Mapping class groups

Let $\pi_0\Diff_{+}(M)$$\pi_0\Diff_{+}(M)$ denote the group of isotopy classes of orientation preserving diffeomorphisms $f\co M \cong M$$f\co M \cong M$ and let $\Aut(H_2(M))$$\Aut(H_2(M))$ be the group of isomorphisms of

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$H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence

(1)$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$

where $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$$H_*(M)$.

• There is an isomphorism $\pi_0\Diff_{+}(S^5) \cong 0$$\pi_0\Diff_{+}(S^5) \cong 0$. By [Cerf1970] and [Smale1962a], $\pi_0\Diff_{+}(S^5) \cong \Theta_6$$\pi_0\Diff_{+}(S^5) \cong \Theta_6$, the group of homotopy $6$$6$-spheres. But by [Kervaire&Milnor1963], $\Theta_6 \cong 0$$\Theta_6 \cong 0$.
• In the homotopy category, $\mathcal{E}_{+}(M)$$\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

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  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ for a general simply-connected 5-manifold in the literature.
  • However if $TH_2(M)$$TH_2(M)$ has no $2$$2$-torsion and no $3$$3$-torsion then $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
  • Even the computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ still leaves an unsolved extension problem in (1) above.

6 References

• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
• [Fang1993] F. Fang, Diffeomorphism groups of simply connected 5-manifolds, unpublished pre-print (1993).

• [Geiges1991] H. Geiges, Contact structures on $1$$1$-connected $5$$5$-manifolds, Mathematika 38 (1991), no.2, 303–311 (1992). MR1147828 (93e:57042) Zbl 0724.57017
• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [Smale1962] S. Smale, On the structure of $5$$5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
• [Stöcker1982] R. Stöcker, On the structure of $5$$5$-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$$(n-1)$-connected $(2n+1)$$(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1964] C. T. C. Wall, Diffeomorphisms of $4$$4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101
• [Zhubr2001] A. V. Zhubr, On a paper of Barden, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), Geom. i Topol. 6, 70–88, 247; translation in J. Math. Sci. (N. Y.) 119 (2004), no. 1, 35–44. MR1846073 (2002e:57040) Zbl 1072.57024

7 External links

• The Wikipedia page on 1-connected 5-manifolds

< k < \infty$, $M_k = M_{\Zz_k}$ is the spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed [[#Constructions_and_examples|above]]. * For \mathcal{M}_{5} be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds

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$M$ and let

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$\mathcal{M}_5^{\text{Spin}}\subset \mathcal{M}_5$ be the subset of diffeomorphism classes of spinable manifolds. The calculation of

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$\mathcal{M}_5^\Spin$ was first obtained by Smale [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden [Barden1965] devised an elegant surgery argument and applied results of [Wall1964] on the diffeomorphism groups of 4-manifolds to give an explicit and complete classification of all of

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$\mathcal{M}_{5}$.

Simply-connected 5-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected 5-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that not every simply-connected 5-dimensional Poincaré space is smoothable. The classification of simply-connected 5-dimensional Poincaré spaces was achieved by Stöcker [Stöcker1982].

2 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$.
• $M_\infty \coloneq S^2 \times S^3$$M_\infty \coloneq S^2 \times S^3$.
• $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, the total space of the non-trivial $S^3$$S^3$-bundle over $S^2$$S^2$.
• $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standard inclusion of $\SO_3 \rightarrow SU_3$$\SO_3 \rightarrow SU_3$.

In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles $M \cong W \cup_f W$$M \cong W \cup_f W$ where $W$$W$ is a certain simply connected $5$$5$-manifold with boundary $\partial W$$\partial W$ a simply-connected $4$$4$-manifold and $f: \partial W \cong \partial W$$f: \partial W \cong \partial W$ is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of $H_2(\partial W)$$H_2(\partial W)$ exist.

2.1 The general spin case

Next we present a construction of simply-connected spin 5-manifolds. A priori the construction depends upon choices, but applying Theorem 4.3 below shows that the choices do not affect the diffeomorphism type of the manifold constructed. Note that all homology groups are with integer coefficients.

Given a finitely generated abelian group $G$$G$, let $X_G$$X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$$H_2(X_G) = G$. The space $X_G$$X_G$ may be realised as a finite CW-complex with only $2$$2$-cells and $3$$3$-cells and so there is an embedding $X_G\to\Rr^6$$X_G\to\Rr^6$. Let

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$N_G$ be a regular neighbourhood of $X_G\subset\Rr^6$$X_G\subset\Rr^6$ and let

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$M_G$ be the boundary of

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$N_G$. Then

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$M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$$H_2(M_G)\cong G \oplus TG$ where $TG$$TG$ is the torsion subgroup of $G$$G$. For example, $M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$$M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$ where $\sharp_r$$\sharp_r$ denotes the $r$$r$-fold connected sum.

2.2 The general non-spin case

For the non-spin case we construct only those manifolds which are boundaries of $6$$6$-manifolds. As in the spin case, the construction depends a priori on choices, but Theorem 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.

Let $(G, w)$$(G, w)$ be a pair with $w\co G \to\Zz_2$$w\co G \to\Zz_2$ a surjective homomorphism and $G$$G$ as above. We shall construct a non-spin 5-manifold $M_{(G, w)}$$M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$$H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$$w_2$ given by $w$$w$ composed with the projection $G \oplus TG \to G$$G \oplus TG \to G$.

If $(G, w) = (\Zz, 1)$$(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$$N_{(\Zz, 1)}$ be the non-trivial $D^4$$D^4$-bundle over $S^2$$S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$$\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$$(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$$N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$$N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$$M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$.

In the general case, present $G = F/i(R)$$G = F/i(R)$ where $i \co R \to F$$i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$$(G, w)$ to $(F, \bar w)$$(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$$F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$$\{r_1, \dots, r_n \}$ is a basis for $R$$R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$$i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$$N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$$N_{(F, \bar w)}$ along spheres representing $i(r_i)$$i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$$M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$$M_{(G, w)}$ is a non-spin manifold as described above.

3 Invariants

Consider the following invariants of a closed simply-connected $5$$5$-manifold

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$M$:

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  $H_2(M)$ be the second integral homology group of

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  $M$,
• $w_2 \co H_2(M) \rightarrow \Zz_2$$w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$$w_2 \in H^2(M; \Zz_2)$,
• $h(M) \in \Nn \cup \{\infty\}$$h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$$r$ such that $2^r \cdot x = 0$$2^r \cdot x = 0$ for some $x \in w_2^{-1}(1)$$x \in w_2^{-1}(1)$. If

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  $M$ is spinable we set $h(M) = 0$$h(M) = 0$.

For example, the manifold $X_{\infty}$$X_{\infty}$ has invariants $H_2(X_{\infty}) \cong \Z$$H_2(X_{\infty}) \cong \Z$, non-trivial $w_2$$w_2$ and $h(X_{\infty}) = \infty$$h(X_{\infty}) = \infty$. The Wu-manifold, $X_{-1}$$X_{-1}$, has invariants $H_2(X_{-1}) = \Zz_2$$H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$$w_2$ and $h(X_{-1}) = 1$$h(X_{-1}) = 1$.

The above list is the minimal list of invariants required to give the classification of closed simply-connected $5$$5$-manifolds: see Theorem 4.2 below.

In addition we mention two further invariants of

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$M$:

• $w_3 \in H^3(M; \Zz_2)$$w_3 \in H^3(M; \Zz_2)$, the third Stiefel-Whitney class,
• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$$b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$$TH_2(M)$, the torsion subgroup of

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  $H_2(M)$.

By [Milnor&Stasheff1974, Problem 8-A], $w_3 = Sq^1(w_2)$$w_3 = Sq^1(w_2)$, and so $w_2$$w_2$ determines $w_3$$w_3$.

By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity $b_M(x, x) = w_2(x)$$b_M(x, x) = w_2(x)$ where we regard $w_2(x)$$w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$$\{0, 1/2\} \subset \Qq/\Zz$. The classification of anti-symmetric linking forms is rather simple and this leads to the fact that one only needs to list the extended natural number $h(M)$$h(M)$ in order to obtain a complete list of invariants of simply-connected $5$$5$-manifolds: This point is clarified in the following sub-section where we report on the classification of anti-symmetric linking forms.

3.1 Linking forms

An abstract non-singular anti-symmetric linking form on a finite abelian group $H$$H$ is a bi-linear function

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$$\displaystyle b \co H \times H \rightarrow \Qq/\Zz$$

such that

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$b(x, y) = 0$ for all

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$y \in H$ if and only if

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$x = 0$ and

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$b(x, y) = -b(y, x)$ for all pairs $x$$x$ and $y$$y$. For example, if $C_k$$C_k$ denotes the cyclic group of order $k$$k$, we have the following linking forms specified by their linking matricies,

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$$\displaystyle b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right),$$

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$$\displaystyle b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),$$

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$$\displaystyle b_j(C_{2^j}) : (C_{2^j} \oplus C_{2^j}) \times (C_{2^j} \oplus C_{2^j}) \to \Qq/\Zz, \quad \left( \begin{array}{cc} \frac{1}{2} & ~\frac{1}{2^j} \\ -\frac{1}{2^j} & ~0 \end{array} \right).$$

If

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$T = \oplus_{i=1}^r C_{k_r}$ is the sum of cyclic groups we shall write

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$b_0(T)$ for the sum

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$\oplus_{i=1}^r b_0(C_{k_r})$.

By [Wall1963, Theorem 3], all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above and indeed such linking forms are classified up to isomorphism by the homomorphism

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$$\displaystyle w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$

Moreover $H$$H$ must be isomorphic to

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$T \oplus T$ or

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$T \oplus T \oplus \Zz_2$ for some finite group $T$$T$ with

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$b(x,x) = 1/2$ if $x$$x$ generates the $\Zz_2$$\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold

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$M$ determines the isomorphism class of the linking form

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$b_M$ and we see that the torsion subgroup of

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$H_2(M)$ is of the form

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$TH_2(M) \cong T \oplus T$ if

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$h(M) \neq 1$ or

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$TH_2(M) \cong T \oplus T \oplus \Zz_2$ if

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$h(M) = 1$ in which case the $\Zz_2$$\Zz_2$ summand is an orthogonal summand of

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$b_M$.

3.2 Values for constructions

The spin manifolds

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$M_G$ all have vanishing $w_2$$w_2$ of course and so by Wall's classification of linking forms we see that the linking form of

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$M_G$ is the linking form $b_0(TG)$$b_0(TG)$.

As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$$M_{(G, \omega)}$ have $w_2$$w_2$ given by projecting to $G$$G$ and then applying $\omega$$\omega$:

$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$ $$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$$

If $M_{(G, \omega)}$$M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$$h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$$b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$$TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$$M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$$b_{M_{(G, \omega)}} = b_0(TG)$.

4 Classification

We first present the most economical classifications of $\mathcal{M}^\Spin_5$$\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$$\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$$G$ with torsion subgroup $TG \cong H \oplus H \oplus C$$TG \cong H \oplus H \oplus C$ where $C$$C$ is trivial or $C \cong \Zz_2$$C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$.

Theorem 4.1 [Smale1962, Theorem p.38]. There is a bijective correspondence

$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$ $$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 4.2 [Barden1965]. The mapping

$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$ $$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$$([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$$[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$$n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

Theorem 4.3 [Barden1965, Theorem 2.2]. Let $M_0$$M_0$ and $M_1$$M_1$ be simply-connected, closed, smooth 5-manifolds and let $A\co H_2(M_0) \cong H_2(M_1)$$A\co H_2(M_0) \cong H_2(M_1)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $A$$A$ is realised by a diffeomorphism $f_A : M_0 \cong M_1$$f_A : M_0 \cong M_1$.

This theorem can re-phrased in categorical language as follows.

• Let $\mathcal{Q}_5$$\mathcal{Q}_5$ be the groupoid with objects $(G, b, w)$$(G, b, w)$ where $G$$G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$$b \co TG \times TG \to \Qq/\Zz$ is an anti-symmetric non-singular linking form and $w\co G \to \Zz_2$$w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$$w(x) = b(x, x)$ for all $x \in TG$$x \in TG$. The morphisms of $\mathcal{Q}_5$$\mathcal{Q}_5$ are isomorphisms of abelian groups commuting with both $w$$w$ and $b$$b$.

• Let $\widetilde{\mathcal{M}}_5$$\widetilde{\mathcal{M}}_5$ be the groupoid with objects simply-connected closed smooth $5$$5$-manifolds embedded in some fixed $\Rr^N$$\Rr^N$ for $N$$N$ large and morphisms isotopy classes of diffeomorphisms.
• Consider the functor

$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$ $$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$$

Theorem 4.4 [Barden1965]. The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$$(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$ is a detecting functor. That is, it induces a bijection on isomorphism classes of objects.

4.1 Enumeration

We first give Barden's enumeration of the set $\mathcal{M}^{}_5$$\mathcal{M}^{}_5$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$$M_\infty\coloneq S^2 \times S^3$, $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$$1 < k < \infty$, $M_k = M_{\Zz_k}$$M_k = M_{\Zz_k}$ is the spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$$H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$$1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$$X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$$H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$ $$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$$-1 \leq j \leq \infty$, $1 < k_i$$1 < k_i$, $k_i$$k_i$ divides $k_{i+1}$$k_{i+1}$ or $k_{i+1} = \infty$$k_{i+1} = \infty$ and $\sharp$$\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$$X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$$X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$$(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5$$\mathcal{M}_5$ as a disjoint union

$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$ $$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$$

where the last two sets denote the diffeomorphism classes of non-spinable 5-manifolds which are respectively boundaries and not boundaries. Then

$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$ $$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$$

5 Further discussion

• As the invariants which classify simply-connected closed oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
• By the construction above every simply-connected closed smooth spinable $5$$5$-manifold embeds into $\Rr^6$$\Rr^6$.

• As the invariants for $-M$$-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

• Barden's results have been nicely discussed and re-proven by Zhubr [Zhubr2001].

5.1 Bordism groups

As $\mathcal{M}_5^\Spin = \{[M_G]\}$$\mathcal{M}_5^\Spin = \{[M_G]\}$,

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$M_G = \partial N_G$ and

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$M_G$ admits a unique spin structure which extends to

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$N_G$ we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group $\Omega_5^\Spin$$\Omega_5^\Spin$ vanishes. The bordism group

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$\Omega_5^{\SO}$ is isomorphic to $\Zz_2$$\Zz_2$, see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number

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$\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

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$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,$$

and

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$w_2(X_{-1}) \neq 0$. It follows that

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$w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and so we have that

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$\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that

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$[X_{-1}]$ is the generator of

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$\Omega_5^{\SO}$ and that a closed, smooth simply-connected 5-manifold

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$M$ is not a boundary if and only if it is diffeomorphic to

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$X_{-1} \sharp M_0$ where $M_0$$M_0$ is a Spin manifold.

5.2 Curvature and contact structures

Every manifold $\sharp_r(S^2 \times S^3)$$\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature by [Boyer&Galicki2006].

The following theorem is an immediate consequence of [Geiges1991, Theorem 1 & Lemma 7].

Theorem 5.1 [Geiges1991].

A simply connected $5$$5$-manifold

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$M$ admits a contact structure if and only if $w_2(M) \in H^2(M; \Zz_2)$$w_2(M) \in H^2(M; \Zz_2)$ has an integral lift in $H^2(M; \Zz)$$H^2(M; \Zz)$. Hence

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$M$ admits a contact structure if and only if $h(M) = 0$$h(M) = 0$ or $\infty$$\infty$; equivalently

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$M$ admits a contact structure if and only if $M \in \mathcal{M}_5^\Spin$$M \in \mathcal{M}_5^\Spin$ or $M \cong M_0 \sharp X_{\infty}$$M \cong M_0 \sharp X_{\infty}$ where $M_0 \in \mathcal{M}_5^\Spin$$M_0 \in \mathcal{M}_5^\Spin$.

Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of $TH_2(M)$$TH_2(M)$ prime to 3 was proven in [Thomas1986].

5.3 Mapping class groups

Let $\pi_0\Diff_{+}(M)$$\pi_0\Diff_{+}(M)$ denote the group of isotopy classes of orientation preserving diffeomorphisms $f\co M \cong M$$f\co M \cong M$ and let $\Aut(H_2(M))$$\Aut(H_2(M))$ be the group of isomorphisms of

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$H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence

(1)$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$

where $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$$H_*(M)$.

• There is an isomphorism $\pi_0\Diff_{+}(S^5) \cong 0$$\pi_0\Diff_{+}(S^5) \cong 0$. By [Cerf1970] and [Smale1962a], $\pi_0\Diff_{+}(S^5) \cong \Theta_6$$\pi_0\Diff_{+}(S^5) \cong \Theta_6$, the group of homotopy $6$$6$-spheres. But by [Kervaire&Milnor1963], $\Theta_6 \cong 0$$\Theta_6 \cong 0$.
• In the homotopy category, $\mathcal{E}_{+}(M)$$\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

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  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ for a general simply-connected 5-manifold in the literature.
  • However if $TH_2(M)$$TH_2(M)$ has no $2$$2$-torsion and no $3$$3$-torsion then $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
  • Even the computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ still leaves an unsolved extension problem in (1) above.

6 References

• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
• [Fang1993] F. Fang, Diffeomorphism groups of simply connected 5-manifolds, unpublished pre-print (1993).

• [Geiges1991] H. Geiges, Contact structures on $1$$1$-connected $5$$5$-manifolds, Mathematika 38 (1991), no.2, 303–311 (1992). MR1147828 (93e:57042) Zbl 0724.57017
• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [Smale1962] S. Smale, On the structure of $5$$5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
• [Stöcker1982] R. Stöcker, On the structure of $5$$5$-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$$(n-1)$-connected $(2n+1)$$(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1964] C. T. C. Wall, Diffeomorphisms of $4$$4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101
• [Zhubr2001] A. V. Zhubr, On a paper of Barden, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), Geom. i Topol. 6, 70–88, 247; translation in J. Math. Sci. (N. Y.) 119 (2004), no. 1, 35–44. MR1846073 (2002e:57040) Zbl 1072.57024

7 External links

• The Wikipedia page on 1-connected 5-manifolds

< j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed [[#Constructions_and_examples|above]] be the non-spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$. With this notation {{cite|Barden1965|Theorem 2.3}} states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by $$ X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$ where $-1 \leq j \leq \infty$, \mathcal{M}_{5} be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds

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$M$ and let

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$\mathcal{M}_5^{\text{Spin}}\subset \mathcal{M}_5$ be the subset of diffeomorphism classes of spinable manifolds. The calculation of

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$\mathcal{M}_5^\Spin$ was first obtained by Smale [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden [Barden1965] devised an elegant surgery argument and applied results of [Wall1964] on the diffeomorphism groups of 4-manifolds to give an explicit and complete classification of all of

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$\mathcal{M}_{5}$.

Simply-connected 5-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected 5-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that not every simply-connected 5-dimensional Poincaré space is smoothable. The classification of simply-connected 5-dimensional Poincaré spaces was achieved by Stöcker [Stöcker1982].

2 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$.
• $M_\infty \coloneq S^2 \times S^3$$M_\infty \coloneq S^2 \times S^3$.
• $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, the total space of the non-trivial $S^3$$S^3$-bundle over $S^2$$S^2$.
• $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standard inclusion of $\SO_3 \rightarrow SU_3$$\SO_3 \rightarrow SU_3$.

In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles $M \cong W \cup_f W$$M \cong W \cup_f W$ where $W$$W$ is a certain simply connected $5$$5$-manifold with boundary $\partial W$$\partial W$ a simply-connected $4$$4$-manifold and $f: \partial W \cong \partial W$$f: \partial W \cong \partial W$ is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of $H_2(\partial W)$$H_2(\partial W)$ exist.

2.1 The general spin case

Next we present a construction of simply-connected spin 5-manifolds. A priori the construction depends upon choices, but applying Theorem 4.3 below shows that the choices do not affect the diffeomorphism type of the manifold constructed. Note that all homology groups are with integer coefficients.

Given a finitely generated abelian group $G$$G$, let $X_G$$X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$$H_2(X_G) = G$. The space $X_G$$X_G$ may be realised as a finite CW-complex with only $2$$2$-cells and $3$$3$-cells and so there is an embedding $X_G\to\Rr^6$$X_G\to\Rr^6$. Let

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$N_G$ be a regular neighbourhood of $X_G\subset\Rr^6$$X_G\subset\Rr^6$ and let

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$M_G$ be the boundary of

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$N_G$. Then

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$M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$$H_2(M_G)\cong G \oplus TG$ where $TG$$TG$ is the torsion subgroup of $G$$G$. For example, $M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$$M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$ where $\sharp_r$$\sharp_r$ denotes the $r$$r$-fold connected sum.

2.2 The general non-spin case

For the non-spin case we construct only those manifolds which are boundaries of $6$$6$-manifolds. As in the spin case, the construction depends a priori on choices, but Theorem 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.

Let $(G, w)$$(G, w)$ be a pair with $w\co G \to\Zz_2$$w\co G \to\Zz_2$ a surjective homomorphism and $G$$G$ as above. We shall construct a non-spin 5-manifold $M_{(G, w)}$$M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$$H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$$w_2$ given by $w$$w$ composed with the projection $G \oplus TG \to G$$G \oplus TG \to G$.

If $(G, w) = (\Zz, 1)$$(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$$N_{(\Zz, 1)}$ be the non-trivial $D^4$$D^4$-bundle over $S^2$$S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$$\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$$(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$$N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$$N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$$M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$.

In the general case, present $G = F/i(R)$$G = F/i(R)$ where $i \co R \to F$$i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$$(G, w)$ to $(F, \bar w)$$(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$$F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$$\{r_1, \dots, r_n \}$ is a basis for $R$$R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$$i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$$N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$$N_{(F, \bar w)}$ along spheres representing $i(r_i)$$i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$$M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$$M_{(G, w)}$ is a non-spin manifold as described above.

3 Invariants

Consider the following invariants of a closed simply-connected $5$$5$-manifold

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$M$:

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  $H_2(M)$ be the second integral homology group of

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  $M$,
• $w_2 \co H_2(M) \rightarrow \Zz_2$$w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$$w_2 \in H^2(M; \Zz_2)$,
• $h(M) \in \Nn \cup \{\infty\}$$h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$$r$ such that $2^r \cdot x = 0$$2^r \cdot x = 0$ for some $x \in w_2^{-1}(1)$$x \in w_2^{-1}(1)$. If

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  $M$ is spinable we set $h(M) = 0$$h(M) = 0$.

For example, the manifold $X_{\infty}$$X_{\infty}$ has invariants $H_2(X_{\infty}) \cong \Z$$H_2(X_{\infty}) \cong \Z$, non-trivial $w_2$$w_2$ and $h(X_{\infty}) = \infty$$h(X_{\infty}) = \infty$. The Wu-manifold, $X_{-1}$$X_{-1}$, has invariants $H_2(X_{-1}) = \Zz_2$$H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$$w_2$ and $h(X_{-1}) = 1$$h(X_{-1}) = 1$.

The above list is the minimal list of invariants required to give the classification of closed simply-connected $5$$5$-manifolds: see Theorem 4.2 below.

In addition we mention two further invariants of

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$M$:

• $w_3 \in H^3(M; \Zz_2)$$w_3 \in H^3(M; \Zz_2)$, the third Stiefel-Whitney class,
• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$$b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$$TH_2(M)$, the torsion subgroup of

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  $H_2(M)$.

By [Milnor&Stasheff1974, Problem 8-A], $w_3 = Sq^1(w_2)$$w_3 = Sq^1(w_2)$, and so $w_2$$w_2$ determines $w_3$$w_3$.

By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity $b_M(x, x) = w_2(x)$$b_M(x, x) = w_2(x)$ where we regard $w_2(x)$$w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$$\{0, 1/2\} \subset \Qq/\Zz$. The classification of anti-symmetric linking forms is rather simple and this leads to the fact that one only needs to list the extended natural number $h(M)$$h(M)$ in order to obtain a complete list of invariants of simply-connected $5$$5$-manifolds: This point is clarified in the following sub-section where we report on the classification of anti-symmetric linking forms.

3.1 Linking forms

An abstract non-singular anti-symmetric linking form on a finite abelian group $H$$H$ is a bi-linear function

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$$\displaystyle b \co H \times H \rightarrow \Qq/\Zz$$

such that

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$b(x, y) = 0$ for all

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$y \in H$ if and only if

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$x = 0$ and

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$b(x, y) = -b(y, x)$ for all pairs $x$$x$ and $y$$y$. For example, if $C_k$$C_k$ denotes the cyclic group of order $k$$k$, we have the following linking forms specified by their linking matricies,

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$$\displaystyle b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right),$$

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$$\displaystyle b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),$$

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$$\displaystyle b_j(C_{2^j}) : (C_{2^j} \oplus C_{2^j}) \times (C_{2^j} \oplus C_{2^j}) \to \Qq/\Zz, \quad \left( \begin{array}{cc} \frac{1}{2} & ~\frac{1}{2^j} \\ -\frac{1}{2^j} & ~0 \end{array} \right).$$

If

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$T = \oplus_{i=1}^r C_{k_r}$ is the sum of cyclic groups we shall write

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$b_0(T)$ for the sum

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$\oplus_{i=1}^r b_0(C_{k_r})$.

By [Wall1963, Theorem 3], all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above and indeed such linking forms are classified up to isomorphism by the homomorphism

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$$\displaystyle w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$

Moreover $H$$H$ must be isomorphic to

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$T \oplus T$ or

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$T \oplus T \oplus \Zz_2$ for some finite group $T$$T$ with

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$b(x,x) = 1/2$ if $x$$x$ generates the $\Zz_2$$\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold

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$M$ determines the isomorphism class of the linking form

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$b_M$ and we see that the torsion subgroup of

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$H_2(M)$ is of the form

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$TH_2(M) \cong T \oplus T$ if

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$h(M) \neq 1$ or

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$TH_2(M) \cong T \oplus T \oplus \Zz_2$ if

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$h(M) = 1$ in which case the $\Zz_2$$\Zz_2$ summand is an orthogonal summand of

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$b_M$.

3.2 Values for constructions

The spin manifolds

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$M_G$ all have vanishing $w_2$$w_2$ of course and so by Wall's classification of linking forms we see that the linking form of

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$M_G$ is the linking form $b_0(TG)$$b_0(TG)$.

As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$$M_{(G, \omega)}$ have $w_2$$w_2$ given by projecting to $G$$G$ and then applying $\omega$$\omega$:

$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$ $$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$$

If $M_{(G, \omega)}$$M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$$h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$$b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$$TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$$M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$$b_{M_{(G, \omega)}} = b_0(TG)$.

4 Classification

We first present the most economical classifications of $\mathcal{M}^\Spin_5$$\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$$\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$$G$ with torsion subgroup $TG \cong H \oplus H \oplus C$$TG \cong H \oplus H \oplus C$ where $C$$C$ is trivial or $C \cong \Zz_2$$C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$.

Theorem 4.1 [Smale1962, Theorem p.38]. There is a bijective correspondence

$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$ $$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 4.2 [Barden1965]. The mapping

$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$ $$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$$([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$$[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$$n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

Theorem 4.3 [Barden1965, Theorem 2.2]. Let $M_0$$M_0$ and $M_1$$M_1$ be simply-connected, closed, smooth 5-manifolds and let $A\co H_2(M_0) \cong H_2(M_1)$$A\co H_2(M_0) \cong H_2(M_1)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $A$$A$ is realised by a diffeomorphism $f_A : M_0 \cong M_1$$f_A : M_0 \cong M_1$.

This theorem can re-phrased in categorical language as follows.

• Let $\mathcal{Q}_5$$\mathcal{Q}_5$ be the groupoid with objects $(G, b, w)$$(G, b, w)$ where $G$$G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$$b \co TG \times TG \to \Qq/\Zz$ is an anti-symmetric non-singular linking form and $w\co G \to \Zz_2$$w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$$w(x) = b(x, x)$ for all $x \in TG$$x \in TG$. The morphisms of $\mathcal{Q}_5$$\mathcal{Q}_5$ are isomorphisms of abelian groups commuting with both $w$$w$ and $b$$b$.

• Let $\widetilde{\mathcal{M}}_5$$\widetilde{\mathcal{M}}_5$ be the groupoid with objects simply-connected closed smooth $5$$5$-manifolds embedded in some fixed $\Rr^N$$\Rr^N$ for $N$$N$ large and morphisms isotopy classes of diffeomorphisms.
• Consider the functor

$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$ $$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$$

Theorem 4.4 [Barden1965]. The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$$(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$ is a detecting functor. That is, it induces a bijection on isomorphism classes of objects.

4.1 Enumeration

We first give Barden's enumeration of the set $\mathcal{M}^{}_5$$\mathcal{M}^{}_5$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$$M_\infty\coloneq S^2 \times S^3$, $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$$1 < k < \infty$, $M_k = M_{\Zz_k}$$M_k = M_{\Zz_k}$ is the spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$$H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$$1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$$X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$$H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$ $$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$$-1 \leq j \leq \infty$, $1 < k_i$$1 < k_i$, $k_i$$k_i$ divides $k_{i+1}$$k_{i+1}$ or $k_{i+1} = \infty$$k_{i+1} = \infty$ and $\sharp$$\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$$X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$$X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$$(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5$$\mathcal{M}_5$ as a disjoint union

$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$ $$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$$

where the last two sets denote the diffeomorphism classes of non-spinable 5-manifolds which are respectively boundaries and not boundaries. Then

$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$ $$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$$

5 Further discussion

• As the invariants which classify simply-connected closed oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
• By the construction above every simply-connected closed smooth spinable $5$$5$-manifold embeds into $\Rr^6$$\Rr^6$.

• As the invariants for $-M$$-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

• Barden's results have been nicely discussed and re-proven by Zhubr [Zhubr2001].

5.1 Bordism groups

As $\mathcal{M}_5^\Spin = \{[M_G]\}$$\mathcal{M}_5^\Spin = \{[M_G]\}$,

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$M_G = \partial N_G$ and

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$M_G$ admits a unique spin structure which extends to

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$N_G$ we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group $\Omega_5^\Spin$$\Omega_5^\Spin$ vanishes. The bordism group

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$\Omega_5^{\SO}$ is isomorphic to $\Zz_2$$\Zz_2$, see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number

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$\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

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$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,$$

and

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$w_2(X_{-1}) \neq 0$. It follows that

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$w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and so we have that

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$\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that

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$[X_{-1}]$ is the generator of

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$\Omega_5^{\SO}$ and that a closed, smooth simply-connected 5-manifold

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$M$ is not a boundary if and only if it is diffeomorphic to

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$X_{-1} \sharp M_0$ where $M_0$$M_0$ is a Spin manifold.

5.2 Curvature and contact structures

Every manifold $\sharp_r(S^2 \times S^3)$$\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature by [Boyer&Galicki2006].

The following theorem is an immediate consequence of [Geiges1991, Theorem 1 & Lemma 7].

Theorem 5.1 [Geiges1991].

A simply connected $5$$5$-manifold

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$M$ admits a contact structure if and only if $w_2(M) \in H^2(M; \Zz_2)$$w_2(M) \in H^2(M; \Zz_2)$ has an integral lift in $H^2(M; \Zz)$$H^2(M; \Zz)$. Hence

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$M$ admits a contact structure if and only if $h(M) = 0$$h(M) = 0$ or $\infty$$\infty$; equivalently

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$M$ admits a contact structure if and only if $M \in \mathcal{M}_5^\Spin$$M \in \mathcal{M}_5^\Spin$ or $M \cong M_0 \sharp X_{\infty}$$M \cong M_0 \sharp X_{\infty}$ where $M_0 \in \mathcal{M}_5^\Spin$$M_0 \in \mathcal{M}_5^\Spin$.

Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of $TH_2(M)$$TH_2(M)$ prime to 3 was proven in [Thomas1986].

5.3 Mapping class groups

Let $\pi_0\Diff_{+}(M)$$\pi_0\Diff_{+}(M)$ denote the group of isotopy classes of orientation preserving diffeomorphisms $f\co M \cong M$$f\co M \cong M$ and let $\Aut(H_2(M))$$\Aut(H_2(M))$ be the group of isomorphisms of

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$H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence

(1)$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$

where $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$$H_*(M)$.

• There is an isomphorism $\pi_0\Diff_{+}(S^5) \cong 0$$\pi_0\Diff_{+}(S^5) \cong 0$. By [Cerf1970] and [Smale1962a], $\pi_0\Diff_{+}(S^5) \cong \Theta_6$$\pi_0\Diff_{+}(S^5) \cong \Theta_6$, the group of homotopy $6$$6$-spheres. But by [Kervaire&Milnor1963], $\Theta_6 \cong 0$$\Theta_6 \cong 0$.
• In the homotopy category, $\mathcal{E}_{+}(M)$$\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

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  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ for a general simply-connected 5-manifold in the literature.
  • However if $TH_2(M)$$TH_2(M)$ has no $2$$2$-torsion and no $3$$3$-torsion then $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
  • Even the computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ still leaves an unsolved extension problem in (1) above.

6 References

• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
• [Fang1993] F. Fang, Diffeomorphism groups of simply connected 5-manifolds, unpublished pre-print (1993).

• [Geiges1991] H. Geiges, Contact structures on $1$$1$-connected $5$$5$-manifolds, Mathematika 38 (1991), no.2, 303–311 (1992). MR1147828 (93e:57042) Zbl 0724.57017
• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [Smale1962] S. Smale, On the structure of $5$$5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
• [Stöcker1982] R. Stöcker, On the structure of $5$$5$-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$$(n-1)$-connected $(2n+1)$$(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1964] C. T. C. Wall, Diffeomorphisms of $4$$4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101
• [Zhubr2001] A. V. Zhubr, On a paper of Barden, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), Geom. i Topol. 6, 70–88, 247; translation in J. Math. Sci. (N. Y.) 119 (2004), no. 1, 35–44. MR1846073 (2002e:57040) Zbl 1072.57024

7 External links

• The Wikipedia page on 1-connected 5-manifolds

< k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the [[Wikipedia:Connected_sum|connected sum]] of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$. An alternative complete enumeration is obtained by writing $\mathcal{M}_5$ as a disjoint union $$ \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$$ where the last two sets denote the diffeomorphism classes of non-spinable 5-manifolds which are respectively boundaries and not boundaries. Then $$ \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$$ == Further discussion == ; * As the invariants which classify simply-connected closed oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism. * By the construction [[#Constructions_and_examples|above]] every simply-connected closed smooth spinable $-manifold embeds into $\Rr^6$. * As the invariants for $-M$ are isomorphic to the invariants of $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly [[amphicheiral]]. * Barden's results have been nicely discussed and re-proven by Zhubr {{cite|Zhubr2001}}. === Bordism groups === ; As $\mathcal{M}_5^\Spin = \{[M_G]\}$, $M_G = \partial N_G$ and $M_G$ admits a unique spin structure which extends to $N_G$ we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the [[wikipedia:Cobordism#Cobordism_classes|bordism group]] $\Omega_5^\Spin$ vanishes. The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$, see for example {{cite|Milnor&Stasheff1974|p 203}}. Moreover this bordism group is detected by the [[wikipedia:Stiefel–Whitney_class#Stiefel.E2.80.93Whitney_numbers|Stiefel-Whitney number]] $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups $$H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,$$ and $w_2(X_{-1}) \neq 0$. It follows that $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and so we have that $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that $[X_{-1}]$ is the generator of $\Omega_5^{\SO}$ and that a closed, smooth simply-connected 5-manifold $M$ is not a boundary if and only if it is diffeomorphic to $X_{-1} \sharp M_0$ where $M_0$ is a Spin manifold. === Curvature and contact structures === ; Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive [[wikipedia:Ricci_curvature|Ricci curvature]] by {{cite|Boyer&Galicki2006}}. The following theorem is an immediate consequence of \cite{Geiges1991|Theorem 1 & Lemma 7}. {{beginthm|Theorem|\cite{Geiges1991}}} A simply connected $-manifold $M$ admits a contact structure if and only if $w_2(M) \in H^2(M; \Zz_2)$ has an integral lift in $H^2(M; \Zz)$. Hence $M$ admits a contact structure if and only if $h(M) = 0$ or $\infty$; equivalently $M$ admits a contact structure if and only if $M \in \mathcal{M}_5^\Spin$ or $M \cong M_0 \sharp X_{\infty}$ where $M_0 \in \mathcal{M}_5^\Spin$. {{endthm}} {{beginrem|Remark}} The special case of this theorem for spin 5-manifolds with the order of $TH_2(M)$ prime to 3 was proven in {{cite|Thomas1986}}. {{endrem}} === Mapping class groups === ; Let $\pi_0\Diff_{+}(M)$ denote the group of [[isotopy]] classes of orientation preserving diffeomorphisms $f\co M \cong M$ and let $\Aut(H_2(M))$ be the group of isomorphisms of $H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying Theorem \ref{thm:Barden-realise} above we obtain the following exact sequence \begin{equation} \label{eq:mcg} 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad \end{equation} where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$. * There is an isomphorism $\pi_0\Diff_{+}(S^5) \cong 0$. By {{cite|Cerf1970}} and {{cite|Smale1962a}}, $\pi_0\Diff_{+}(S^5) \cong \Theta_6$, the group of [[wikipedia:Homotopy_sphere|homotopy $-spheres]]. But by {{cite|Kervaire&Milnor1963}}, $\Theta_6 \cong 0$. * In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of $M$, has been extensively investigated by {{cite|Baues&Buth1996}} and is already seen to be relatively complex. * Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold in the literature. ** However if $TH_2(M)$ has no $-torsion and no $-torsion then $\pi_0\SDiff(M)$ was computed in \cite{Fang1993}. This computation agrees with a more recent conjectured answer: please see the [[Talk:1-connected 5-manifolds#Conjecture about the mapping class group of 1-connected 5-manifolds|discussion page]]. ** Even the computation of $\pi_0\SDiff(M)$ still leaves an unsolved extension problem in (\ref{eq:mcg}) above. == References == {{#RefList:}} == External links == * The Wikipedia page on [[Wikipedia:5-manifold|1-connected 5-manifolds]] [[Category:Manifolds]] [[Category:Highly-connected manifolds]]\mathcal{M}_{5} be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds

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$M$ and let

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$\mathcal{M}_5^{\text{Spin}}\subset \mathcal{M}_5$ be the subset of diffeomorphism classes of spinable manifolds. The calculation of

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$\mathcal{M}_5^\Spin$ was first obtained by Smale [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden [Barden1965] devised an elegant surgery argument and applied results of [Wall1964] on the diffeomorphism groups of 4-manifolds to give an explicit and complete classification of all of

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$\mathcal{M}_{5}$.

Simply-connected 5-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected 5-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that not every simply-connected 5-dimensional Poincaré space is smoothable. The classification of simply-connected 5-dimensional Poincaré spaces was achieved by Stöcker [Stöcker1982].

2 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$.
• $M_\infty \coloneq S^2 \times S^3$$M_\infty \coloneq S^2 \times S^3$.
• $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, the total space of the non-trivial $S^3$$S^3$-bundle over $S^2$$S^2$.
• $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standard inclusion of $\SO_3 \rightarrow SU_3$$\SO_3 \rightarrow SU_3$.

In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles $M \cong W \cup_f W$$M \cong W \cup_f W$ where $W$$W$ is a certain simply connected $5$$5$-manifold with boundary $\partial W$$\partial W$ a simply-connected $4$$4$-manifold and $f: \partial W \cong \partial W$$f: \partial W \cong \partial W$ is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of $H_2(\partial W)$$H_2(\partial W)$ exist.

2.1 The general spin case

Next we present a construction of simply-connected spin 5-manifolds. A priori the construction depends upon choices, but applying Theorem 4.3 below shows that the choices do not affect the diffeomorphism type of the manifold constructed. Note that all homology groups are with integer coefficients.

Given a finitely generated abelian group $G$$G$, let $X_G$$X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$$H_2(X_G) = G$. The space $X_G$$X_G$ may be realised as a finite CW-complex with only $2$$2$-cells and $3$$3$-cells and so there is an embedding $X_G\to\Rr^6$$X_G\to\Rr^6$. Let

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$N_G$ be a regular neighbourhood of $X_G\subset\Rr^6$$X_G\subset\Rr^6$ and let

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$M_G$ be the boundary of

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$N_G$. Then

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$M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$$H_2(M_G)\cong G \oplus TG$ where $TG$$TG$ is the torsion subgroup of $G$$G$. For example, $M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$$M_{\Zz^r} \cong \sharp_r (S^2 \times S^3)$ where $\sharp_r$$\sharp_r$ denotes the $r$$r$-fold connected sum.

2.2 The general non-spin case

For the non-spin case we construct only those manifolds which are boundaries of $6$$6$-manifolds. As in the spin case, the construction depends a priori on choices, but Theorem 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.

Let $(G, w)$$(G, w)$ be a pair with $w\co G \to\Zz_2$$w\co G \to\Zz_2$ a surjective homomorphism and $G$$G$ as above. We shall construct a non-spin 5-manifold $M_{(G, w)}$$M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$$H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$$w_2$ given by $w$$w$ composed with the projection $G \oplus TG \to G$$G \oplus TG \to G$.

If $(G, w) = (\Zz, 1)$$(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$$N_{(\Zz, 1)}$ be the non-trivial $D^4$$D^4$-bundle over $S^2$$S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$$\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$$(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$$N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$$N_{(\Zz, 1)} \natural_r (S^2 \times D^4)$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$$M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$.

In the general case, present $G = F/i(R)$$G = F/i(R)$ where $i \co R \to F$$i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$$(G, w)$ to $(F, \bar w)$$(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$$F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$$\{r_1, \dots, r_n \}$ is a basis for $R$$R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$$i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$$N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$$N_{(F, \bar w)}$ along spheres representing $i(r_i)$$i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$$M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$$M_{(G, w)}$ is a non-spin manifold as described above.

3 Invariants

Consider the following invariants of a closed simply-connected $5$$5$-manifold

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$M$:

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  $H_2(M)$ be the second integral homology group of

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  $M$,
• $w_2 \co H_2(M) \rightarrow \Zz_2$$w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$$w_2 \in H^2(M; \Zz_2)$,
• $h(M) \in \Nn \cup \{\infty\}$$h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$$r$ such that $2^r \cdot x = 0$$2^r \cdot x = 0$ for some $x \in w_2^{-1}(1)$$x \in w_2^{-1}(1)$. If

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  $M$ is spinable we set $h(M) = 0$$h(M) = 0$.

For example, the manifold $X_{\infty}$$X_{\infty}$ has invariants $H_2(X_{\infty}) \cong \Z$$H_2(X_{\infty}) \cong \Z$, non-trivial $w_2$$w_2$ and $h(X_{\infty}) = \infty$$h(X_{\infty}) = \infty$. The Wu-manifold, $X_{-1}$$X_{-1}$, has invariants $H_2(X_{-1}) = \Zz_2$$H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$$w_2$ and $h(X_{-1}) = 1$$h(X_{-1}) = 1$.

The above list is the minimal list of invariants required to give the classification of closed simply-connected $5$$5$-manifolds: see Theorem 4.2 below.

In addition we mention two further invariants of

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$M$:

• $w_3 \in H^3(M; \Zz_2)$$w_3 \in H^3(M; \Zz_2)$, the third Stiefel-Whitney class,
• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$$b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$$TH_2(M)$, the torsion subgroup of

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  $H_2(M)$.

By [Milnor&Stasheff1974, Problem 8-A], $w_3 = Sq^1(w_2)$$w_3 = Sq^1(w_2)$, and so $w_2$$w_2$ determines $w_3$$w_3$.

By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity $b_M(x, x) = w_2(x)$$b_M(x, x) = w_2(x)$ where we regard $w_2(x)$$w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$$\{0, 1/2\} \subset \Qq/\Zz$. The classification of anti-symmetric linking forms is rather simple and this leads to the fact that one only needs to list the extended natural number $h(M)$$h(M)$ in order to obtain a complete list of invariants of simply-connected $5$$5$-manifolds: This point is clarified in the following sub-section where we report on the classification of anti-symmetric linking forms.

3.1 Linking forms

An abstract non-singular anti-symmetric linking form on a finite abelian group $H$$H$ is a bi-linear function

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$$\displaystyle b \co H \times H \rightarrow \Qq/\Zz$$

such that

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$b(x, y) = 0$ for all

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$y \in H$ if and only if

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$x = 0$ and

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$b(x, y) = -b(y, x)$ for all pairs $x$$x$ and $y$$y$. For example, if $C_k$$C_k$ denotes the cyclic group of order $k$$k$, we have the following linking forms specified by their linking matricies,

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$$\displaystyle b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right),$$

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$$\displaystyle b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),$$

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$$\displaystyle b_j(C_{2^j}) : (C_{2^j} \oplus C_{2^j}) \times (C_{2^j} \oplus C_{2^j}) \to \Qq/\Zz, \quad \left( \begin{array}{cc} \frac{1}{2} & ~\frac{1}{2^j} \\ -\frac{1}{2^j} & ~0 \end{array} \right).$$

If

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$T = \oplus_{i=1}^r C_{k_r}$ is the sum of cyclic groups we shall write

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$b_0(T)$ for the sum

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$\oplus_{i=1}^r b_0(C_{k_r})$.

By [Wall1963, Theorem 3], all non-singular anti-symmetric linking forms are isomorphic to a sum of the linking forms above and indeed such linking forms are classified up to isomorphism by the homomorphism

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$$\displaystyle w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$

Moreover $H$$H$ must be isomorphic to

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$T \oplus T$ or

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$T \oplus T \oplus \Zz_2$ for some finite group $T$$T$ with

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$b(x,x) = 1/2$ if $x$$x$ generates the $\Zz_2$$\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold

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$M$ determines the isomorphism class of the linking form

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$b_M$ and we see that the torsion subgroup of

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$H_2(M)$ is of the form

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$TH_2(M) \cong T \oplus T$ if

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$h(M) \neq 1$ or

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$TH_2(M) \cong T \oplus T \oplus \Zz_2$ if

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$h(M) = 1$ in which case the $\Zz_2$$\Zz_2$ summand is an orthogonal summand of

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$b_M$.

3.2 Values for constructions

The spin manifolds

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$M_G$ all have vanishing $w_2$$w_2$ of course and so by Wall's classification of linking forms we see that the linking form of

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$M_G$ is the linking form $b_0(TG)$$b_0(TG)$.

As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$$M_{(G, \omega)}$ have $w_2$$w_2$ given by projecting to $G$$G$ and then applying $\omega$$\omega$:

$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$ $$\displaystyle w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$$

If $M_{(G, \omega)}$$M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$$h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$$b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$$TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$$M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$$b_{M_{(G, \omega)}} = b_0(TG)$.

4 Classification

We first present the most economical classifications of $\mathcal{M}^\Spin_5$$\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$$\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$$G$ with torsion subgroup $TG \cong H \oplus H \oplus C$$TG \cong H \oplus H \oplus C$ where $C$$C$ is trivial or $C \cong \Zz_2$$C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$${\mathcal Ab}^{T \oplus T \oplus *}$.

Theorem 4.1 [Smale1962, Theorem p.38]. There is a bijective correspondence

$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$ $$\displaystyle \mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 4.2 [Barden1965]. The mapping

$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$ $$\displaystyle \mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$$([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$$[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$$n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

Theorem 4.3 [Barden1965, Theorem 2.2]. Let $M_0$$M_0$ and $M_1$$M_1$ be simply-connected, closed, smooth 5-manifolds and let $A\co H_2(M_0) \cong H_2(M_1)$$A\co H_2(M_0) \cong H_2(M_1)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $A$$A$ is realised by a diffeomorphism $f_A : M_0 \cong M_1$$f_A : M_0 \cong M_1$.

This theorem can re-phrased in categorical language as follows.

• Let $\mathcal{Q}_5$$\mathcal{Q}_5$ be the groupoid with objects $(G, b, w)$$(G, b, w)$ where $G$$G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$$b \co TG \times TG \to \Qq/\Zz$ is an anti-symmetric non-singular linking form and $w\co G \to \Zz_2$$w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$$w(x) = b(x, x)$ for all $x \in TG$$x \in TG$. The morphisms of $\mathcal{Q}_5$$\mathcal{Q}_5$ are isomorphisms of abelian groups commuting with both $w$$w$ and $b$$b$.

• Let $\widetilde{\mathcal{M}}_5$$\widetilde{\mathcal{M}}_5$ be the groupoid with objects simply-connected closed smooth $5$$5$-manifolds embedded in some fixed $\Rr^N$$\Rr^N$ for $N$$N$ large and morphisms isotopy classes of diffeomorphisms.
• Consider the functor

$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$ $$\displaystyle (b, w_2) \co \widetilde{\mathcal{M}}_5\to \mathcal{Q}_5:~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$$

Theorem 4.4 [Barden1965]. The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$$(b, w_2)\co \widetilde{\mathcal{M}}_5 \to \mathcal{Q}_5$ is a detecting functor. That is, it induces a bijection on isomorphism classes of objects.

4.1 Enumeration

We first give Barden's enumeration of the set $\mathcal{M}^{}_5$$\mathcal{M}^{}_5$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$$X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$$M_\infty\coloneq S^2 \times S^3$, $X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$$X_\infty \coloneq S^2 \tilde \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$$X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$$1 < k < \infty$, $M_k = M_{\Zz_k}$$M_k = M_{\Zz_k}$ is the spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$$H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$$1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$$X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$$H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$ $$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$$-1 \leq j \leq \infty$, $1 < k_i$$1 < k_i$, $k_i$$k_i$ divides $k_{i+1}$$k_{i+1}$ or $k_{i+1} = \infty$$k_{i+1} = \infty$ and $\sharp$$\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$$X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$$X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$$(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5$$\mathcal{M}_5$ as a disjoint union

$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$ $$\displaystyle \mathcal{M}_5 = \mathcal{M}_5^\Spin \sqcup \mathcal{M}_5^{w_2, = \partial} \sqcup \mathcal{M}_5^{w_2,\neq \partial}$$

where the last two sets denote the diffeomorphism classes of non-spinable 5-manifolds which are respectively boundaries and not boundaries. Then

$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$ $$\displaystyle \mathcal{M}_5^\Spin = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial} = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial} = \{ [ X_{-1} \sharp M_G] \}.$$

5 Further discussion

• As the invariants which classify simply-connected closed oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
• By the construction above every simply-connected closed smooth spinable $5$$5$-manifold embeds into $\Rr^6$$\Rr^6$.

• As the invariants for $-M$$-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

• Barden's results have been nicely discussed and re-proven by Zhubr [Zhubr2001].

5.1 Bordism groups

As $\mathcal{M}_5^\Spin = \{[M_G]\}$$\mathcal{M}_5^\Spin = \{[M_G]\}$,

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$M_G = \partial N_G$ and

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$M_G$ admits a unique spin structure which extends to

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$N_G$ we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group $\Omega_5^\Spin$$\Omega_5^\Spin$ vanishes. The bordism group

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$\Omega_5^{\SO}$ is isomorphic to $\Zz_2$$\Zz_2$, see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number

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$\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

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$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,$$

and

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$w_2(X_{-1}) \neq 0$. It follows that

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$w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and so we have that

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$\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that

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$[X_{-1}]$ is the generator of

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$\Omega_5^{\SO}$ and that a closed, smooth simply-connected 5-manifold

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$M$ is not a boundary if and only if it is diffeomorphic to

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$X_{-1} \sharp M_0$ where $M_0$$M_0$ is a Spin manifold.

5.2 Curvature and contact structures

Every manifold $\sharp_r(S^2 \times S^3)$$\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature by [Boyer&Galicki2006].

The following theorem is an immediate consequence of [Geiges1991, Theorem 1 & Lemma 7].

Theorem 5.1 [Geiges1991].

A simply connected $5$$5$-manifold

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$M$ admits a contact structure if and only if $w_2(M) \in H^2(M; \Zz_2)$$w_2(M) \in H^2(M; \Zz_2)$ has an integral lift in $H^2(M; \Zz)$$H^2(M; \Zz)$. Hence

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$M$ admits a contact structure if and only if $h(M) = 0$$h(M) = 0$ or $\infty$$\infty$; equivalently

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$M$ admits a contact structure if and only if $M \in \mathcal{M}_5^\Spin$$M \in \mathcal{M}_5^\Spin$ or $M \cong M_0 \sharp X_{\infty}$$M \cong M_0 \sharp X_{\infty}$ where $M_0 \in \mathcal{M}_5^\Spin$$M_0 \in \mathcal{M}_5^\Spin$.

Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of $TH_2(M)$$TH_2(M)$ prime to 3 was proven in [Thomas1986].

5.3 Mapping class groups

Let $\pi_0\Diff_{+}(M)$$\pi_0\Diff_{+}(M)$ denote the group of isotopy classes of orientation preserving diffeomorphisms $f\co M \cong M$$f\co M \cong M$ and let $\Aut(H_2(M))$$\Aut(H_2(M))$ be the group of isomorphisms of

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$H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence

(1)$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad$

where $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$$H_*(M)$.

• There is an isomphorism $\pi_0\Diff_{+}(S^5) \cong 0$$\pi_0\Diff_{+}(S^5) \cong 0$. By [Cerf1970] and [Smale1962a], $\pi_0\Diff_{+}(S^5) \cong \Theta_6$$\pi_0\Diff_{+}(S^5) \cong \Theta_6$, the group of homotopy $6$$6$-spheres. But by [Kervaire&Milnor1963], $\Theta_6 \cong 0$$\Theta_6 \cong 0$.
• In the homotopy category, $\mathcal{E}_{+}(M)$$\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

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  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ for a general simply-connected 5-manifold in the literature.
  • However if $TH_2(M)$$TH_2(M)$ has no $2$$2$-torsion and no $3$$3$-torsion then $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
  • Even the computation of $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ still leaves an unsolved extension problem in (1) above.

6 References

• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
• [Fang1993] F. Fang, Diffeomorphism groups of simply connected 5-manifolds, unpublished pre-print (1993).

• [Geiges1991] H. Geiges, Contact structures on $1$$1$-connected $5$$5$-manifolds, Mathematika 38 (1991), no.2, 303–311 (1992). MR1147828 (93e:57042) Zbl 0724.57017
• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [Smale1962] S. Smale, On the structure of $5$$5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
• [Stöcker1982] R. Stöcker, On the structure of $5$$5$-dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$$(n-1)$-connected $(2n+1)$$(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
• [Wall1964] C. T. C. Wall, Diffeomorphisms of $4$$4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101
• [Zhubr2001] A. V. Zhubr, On a paper of Barden, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), Geom. i Topol. 6, 70–88, 247; translation in J. Math. Sci. (N. Y.) 119 (2004), no. 1, 35–44. MR1846073 (2002e:57040) Zbl 1072.57024

7 External links

• The Wikipedia page on 1-connected 5-manifolds