5-manifolds: 1-connected

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An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print.

You may view the version used for publication as of 09:22, 1 April 2011 and the changes since publication.

1 Introduction

Let $== Introduction == <wikitex>; Let $\mathcal{M}_{5}(e)$ 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 $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ be the subset of diffeomorphism classes of [[wikipedia:Spin_manifold|spinable manifolds]]. The calculation of $\mathcal{M}_5^{\Spin}(e)$ was first obtained by Smale in {{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 $\mathcal{M}_{5}(e)$. 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 may be found in {{cite|Stöcker1982}}. </wikitex> == Constructions and examples== <wikitex>; We first list some familiar 5-manifolds using Barden's notation: * $X_0 \coloneq S^5$. * $M_\infty \coloneq S^2 \times S^3$. * $X_\infty \coloneq S^2 \times_{\gamma} S^3$, 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 standadard inclusion of $\SO_3 \rightarrow SU_3$. * Next we present a construction of Spin 5-manifolds. 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 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. * For the non-Spin case 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 [[#Invariants|second 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 a 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. * 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.  </wikitex> == Invariants == <wikitex>; Consider the following invariants of a closed simply-connected 5-manifold $M$. * $H_2(M)$ be the second integral homology group of $M$, with torsion subgroup $TH_2(M)$. * $w_2 \co H_2(M) \rightarrow \Zz_2$, 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 $x^{2^r} = e$ and $x \in w_2^{-1}(1)$. If $M$ is Spin we set $h(M) = 0$. * $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the [[wikipedia:Poincaré duality#Bilinear_pairings_formulation|linking form]] of $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$. 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 $\{0, 1/2\} \subset \Qq/\Zz$. For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$. An abstract non-singular, anti-symmetric linking form $b \co H \times H \rightarrow \Qq/\Zz$ on a finite group $H$ is a bi-linear function such that $b(x, y) = -b(y, x)$ and $b(x, y) = 0$ for all $y \in H$ if and only if $x = 0$. By {{cite|Wall1963}} such linking forms are classified up to isomorphism by the homomorphism $H \rightarrow \Zz_2, x \mapsto b(x, x)$. 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> == Classification == <wikitex>; We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. 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. {{beginthm|Theorem|{{cite|Smale1962|}}}} There is a bijective correspondence $$\mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$ {{endthm}} {{beginthm|Theorem|{{cite|Barden1965}}}} The mapping $$\mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$ 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}}}} 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. {{endthm}} This theorem can re-phrased in categorical language as follows. * Let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a 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 $\mathcal{Q}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$. * Let $\widetilde{\mathcal{M}}_5(e)$ be a small groupoid with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy classes of diffeomorphisms.  * Consider the functor $$(b, w_2) \co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e):~~~ M \mapsto (H_2(M), b_M, w_2(M)), ~~~ f \co M_0 \cong M_1 \mapsto H_2(f).$$ {{beginthm|Theorem|{{cite|Barden1965}}}} The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$ is a detecting functor. That is, it is surjective on isomorphism classes of objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$. {{endthm}} </wikitex> === Enumeration === <wikitex>; We first give Barden's enumeration of the set $\mathcal{M}^{}_5(e)$, {{cite|Barden1965|Theorem 2.3}}. * $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$. * For \mathcal{M}_{5}(e)$ be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds $M$ and let $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ be the subset of diffeomorphism classes of spinable manifolds. The calculation of $\mathcal{M}_5^{\Spin}(e)$ was first obtained by Smale in [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 $\mathcal{M}_{5}(e)$ .

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 may be found in [Stöcker1982].

2 Constructions and examples

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

$X_0 \coloneq S^5$ .
$M_\infty \coloneq S^2 \times S^3$ .
$X_\infty \coloneq S^2 \times_{\gamma} S^3$ , 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 standadard inclusion of $\SO_3 \rightarrow SU_3$ .

Next we present a construction of Spin 5-manifolds. 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 and so there is an embedding $X_G\to\Rr^6$ . Let $N(G)$ be a 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.

For the non-Spin case 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 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 a 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.

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

3 Invariants

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

$H_2(M)$ be the second integral homology group of $M$ , with torsion subgroup $TH_2(M)$ .
$w_2 \co H_2(M) \rightarrow \Zz_2$ , the homomorphism defined by evaluation with the second 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 $x^{2^r} = e$ and $x \in w_2^{-1}(1)$ . If $M$ is Spin we set $h(M) = 0$ .

$b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$ , the linking form of $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$ .

By [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 $\{0, 1/2\} \subset \Qq/\Zz$ .

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

An abstract non-singular, anti-symmetric linking form $b \co H \times H \rightarrow \Qq/\Zz$ on a finite group $H$ is a bi-linear function such that $b(x, y) = -b(y, x)$ and $b(x, y) = 0$ for all $y \in H$ if and only if $x = 0$ . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism $H \rightarrow \Zz_2, x \mapsto b(x, x)$ . 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 $b_M$ .

4 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$ . 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.

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

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

Theorem 4.2 [Barden1965]. The mapping

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

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 $n = 1$ .

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

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.

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

Let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a 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 $\mathcal{Q}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$ .

Let $\widetilde{\mathcal{M}}_5(e)$ be a small groupoid with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy classes of diffeomorphisms.
Consider the functor

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

The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$ is a detecting functor. That is, it is surjective on isomorphism classes of objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$ .

4.1 Enumeration

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

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

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 $-1 \leq j \leq \infty$ , $1 < k_i$ , $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the 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 $\mathcal{M}_5(e)$ as a disjoint union

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

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

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

5 Further discussion

As the invariants which classify 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, Spin 5-manifold embedds 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 are nicely discussed and re-proven in [Zhubr2001].

5.1 Bordism groups

As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group $\Omega_5^{\Spin}$ vanishes.

The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$ . The Wu-manifold has cohomology groups

$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$ $\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$

and $w_2(X_{-1}) \neq 0$ $w_2(X_{-1}) \neq 0$ . It follows that $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and that $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$ $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$ . We see that $[X_{-1}]$ $[X_{-1}]$ is the generator of $\Omega_5^{\SO}$ $\Omega_5^{\SO}$ and that a closed, smooth 5-manifold $M$ $M$ is not a boundary if and only if it is diffeomorphic to $X_{-1} \sharp M_0$ $X_{-1} \sharp M_0$ where $M_0$ $M_0$ is a Spin manifold.

5.2 Curvature and contact structures

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

Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a contact structure by [Thomas1986].

5.3 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 the second statement of Barden's classification above we obtain an exact sequence

$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$ $\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$

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 [Cerf1970] and [Smale1962a], $\pi_0\Diff_{+}(S^5) \cong \Theta_6$ , the group of homotopy $6$ -spheres. But by [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 [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, there is a conjectured answer: please see the discussion page.

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
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< 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}(e) be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds $M$ $M$ and let $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ be the subset of diffeomorphism classes of spinable manifolds. The calculation of $\mathcal{M}_5^{\Spin}(e)$ $\mathcal{M}_5^{\Spin}(e)$ was first obtained by Smale in [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 $\mathcal{M}_{5}(e)$ $\mathcal{M}_{5}(e)$ .

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 may be found in [Stöcker1982].

2 Constructions and examples

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

$X_0 \coloneq S^5$ .
$M_\infty \coloneq S^2 \times S^3$ .
$X_\infty \coloneq S^2 \times_{\gamma} S^3$ , 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 standadard inclusion of $\SO_3 \rightarrow SU_3$ .

Next we present a construction of Spin 5-manifolds. 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 and so there is an embedding $X_G\to\Rr^6$ . Let $N(G)$ be a 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.

For the non-Spin case 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 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 a 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.

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

3 Invariants

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

$H_2(M)$ be the second integral homology group of $M$ , with torsion subgroup $TH_2(M)$ .
$w_2 \co H_2(M) \rightarrow \Zz_2$ , the homomorphism defined by evaluation with the second 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 $x^{2^r} = e$ and $x \in w_2^{-1}(1)$ . If $M$ is Spin we set $h(M) = 0$ .

$b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$ , the linking form of $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$ .

By [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 $\{0, 1/2\} \subset \Qq/\Zz$ .

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

An abstract non-singular, anti-symmetric linking form $b \co H \times H \rightarrow \Qq/\Zz$ on a finite group $H$ is a bi-linear function such that $b(x, y) = -b(y, x)$ and $b(x, y) = 0$ for all $y \in H$ if and only if $x = 0$ . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism $H \rightarrow \Zz_2, x \mapsto b(x, x)$ . 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 $b_M$ .

4 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$ . 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.

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

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

Theorem 4.2 [Barden1965]. The mapping

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

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 $n = 1$ .

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

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.

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

Let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a 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 $\mathcal{Q}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$ .

Let $\widetilde{\mathcal{M}}_5(e)$ be a small groupoid with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy classes of diffeomorphisms.
Consider the functor

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

The functor $(b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$ is a detecting functor. That is, it is surjective on isomorphism classes of objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$ .