5-manifolds: 1-connected
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− | == Introduction == | + | {{Authors|Diarmuid Crowley}} |
− | <wikitex>; | + | ==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 $\mathcal{M}_{5}$. | 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 $\mathcal{M}_{5}$. | ||
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=== The general spin case === | === The general spin case === | ||
<wikitex>; | <wikitex>; | ||
− | Next we present a construction of simply-connected 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 with only $2$-cells and $3$-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. | + | 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 $2$-cells and $3$-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> | </wikitex> | ||
=== The general non-spin case === | === The general non-spin case === | ||
<wikitex>; | <wikitex>; | ||
− | For the non-spin case we construct only those manifolds which are boundaries of $6$-manifolds. 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 $G \oplus TG \to G$. | + | For the non-spin case we construct only those manifolds which are boundaries of $6$-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 $M_{(G, w)} = X_{\infty} \sharp_r (S^2 \times S^3)$. | 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 | + | 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> | </wikitex> | ||
== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | Consider the following invariants of a closed simply-connected 5-manifold $M$ | + | Consider the following invariants of a closed simply-connected $5$-manifold $M$: |
− | * $H_2(M)$ be the second integral homology group of $M$, | + | * $H_2(M)$ 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 $M$, $w_2 \in H^2(M; \Zz_2)$ | + | * $w_2 \co H_2(M) \rightarrow \Zz_2$, 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 $ | + | * $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that $2^r \cdot x = 0$ 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 $5$-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 $5$-manifolds: This point is clarified in the following sub-section | ||
+ | where we report on the classification of anti-symmetric linking forms. | ||
</wikitex> | </wikitex> | ||
=== Linking forms === | === Linking forms === | ||
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An abstract non-singular anti-symmetric linking form on a finite abelian group $H$ is a bi-linear function | An abstract non-singular anti-symmetric linking form on a finite abelian group $H$ is a bi-linear function | ||
$$ b \co H \times H \rightarrow \Qq/\Zz $$ | $$ b \co H \times H \rightarrow \Qq/\Zz $$ | ||
− | 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, we have the following linking forms specified by their linking matricies | + | 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, |
$$ b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right), $$ | $$ b_{-1} : C_2 \times C_2 \to \Qq/\Zz, \quad \left( \frac{1}{2} \right), $$ | ||
$$ b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),$$ | $$ b_0(C_k) : (C_k \oplus C_k) \times (C_k \oplus C_k) \to \Qq/\Zz, \quad \left( \begin{array}{cc} 0 & ~\frac{1}{k} \\-\frac{1}{k} & ~0 \end{array} \right),$$ | ||
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If $T = \oplus_{i=1}^r C_{k_r}$ is the sum of cyclic groups we shall write $b_0(T)$ for the sum $\oplus_{i=1}^r b_0(C_{k_r})$. | 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 | + | 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 |
$$ w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$ | $$ w(b) : H \rightarrow \Zz_2, \quad x \mapsto b(x, x).$$ | ||
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 $b_M$. | 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 $b_M$. | ||
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As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$ have $w_2$ given by projecting to $G$ and then applying $\omega$: | As we mentioned above, the non-spin manifolds $M_{(G, \omega)}$ have $w_2$ given by projecting to $G$ and then applying $\omega$: | ||
− | $$ w_2 = \omega \circ pr : TG \oplus G \to G \to \ | + | $$ w_2 = \omega \circ pr : TG \oplus G \to G \to \Zz_2.$$ |
If $M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$. | If $M_{(G, \omega)}$ has height finite height $h(M_{(G, \omega)}) = j$ then it follows from Wall's classification of linking forms that $b_{M_{(G, \omega)}} \cong b_1(C_{2^j}) \oplus b_0(T)$ where $TG \cong C_{2^j} \oplus T$ and if $M_{(G, \omega)}$ has infinite height then $b_{M_{(G, \omega)}} = b_0(TG)$. | ||
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We first present the most economical classifications of $\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$. | We first present the most economical classifications of $\mathcal{M}^\Spin_5$ and $\mathcal{M}_5$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets of ${\mathcal Ab}^{T \oplus T \oplus *}$. | ||
− | {{beginthm|Theorem|{{cite|Smale1962|}}}} There is a bijective correspondence $$\mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$ | + | {{beginthm|Theorem|{{cite|Smale1962|Theorem p.38}}}} There is a bijective correspondence $$\mathcal{M}_5^\Spin \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$ |
{{endthm}} | {{endthm}} | ||
− | {{beginthm|Theorem|{{cite|Barden1965}}}} The mapping | + | {{beginthm|Theorem|{{cite|Barden1965}}}} \label{thm:minimal_classification} The mapping |
$$\mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) | $$\mathcal{M}_{5} \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) | ||
, \quad [M] \mapsto ([H_2(M)], h(M))$$ | , \quad [M] \mapsto ([H_2(M)], h(M))$$ | ||
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* The classification of simply-connected Spin 5-manifolds was one of the first applications of the [[wikipedia:H-cobordism|h-cobordism theorem]]. | * The classification of simply-connected Spin 5-manifolds was one of the first applications of the [[wikipedia:H-cobordism|h-cobordism theorem]]. | ||
--> | --> | ||
− | * By the construction [[# | + | * By the construction [[#Constructions_and_examples|above]] every simply-connected closed smooth spinable $5$-manifold embeds into $\Rr^6$. |
* As the invariants for $-M$ are isomorphic to the invariants of $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly [[amphicheiral]]. | * As the invariants for $-M$ are isomorphic to the invariants of $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly [[amphicheiral]]. | ||
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The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$, see for example {{cite|Milnor&Stasheff1974|p 203}}. Moreover this bordism group is detected by the [[wikipedia:Stiefel–Whitney_class#Stiefel.E2.80.93Whitney_numbers|Stiefel-Whitney number]] $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups | The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$, see for example {{cite|Milnor&Stasheff1974|p 203}}. Moreover this bordism group is detected by the [[wikipedia:Stiefel–Whitney_class#Stiefel.E2.80.93Whitney_numbers|Stiefel-Whitney number]] $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups | ||
− | $$H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2, \quad \ast = 0, 1, 2, 3, 4, 5,$$ and $w_2(X_{-1}) \neq 0$. It follows that $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and so we have that $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that $[X_{-1}]$ is the generator of $\Omega_5^{\SO}$ and that a closed, smooth 5-manifold $M$ is not a boundary if and only if it is diffeomorphic to $X_{-1} \sharp M_0$ where $M_0$ is a Spin manifold. | + | $$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. |
</wikitex> | </wikitex> | ||
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Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive [[wikipedia:Ricci_curvature|Ricci curvature]] by {{cite|Boyer&Galicki2006}}. | 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 $5$-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}} | ||
</wikitex> | </wikitex> | ||
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<wikitex>; | <wikitex>; | ||
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 | 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 $H_*(M)$. | 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 $6$-spheres]]. But by {{cite|Kervaire&Milnor1963}}, $\Theta_6 \cong 0$. | * 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 $6$-spheres]]. But by {{cite|Kervaire&Milnor1963}}, $\Theta_6 \cong 0$. | ||
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* Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold in the literature. | * 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 $2$-torsion and no $3$-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]]. | ** However if $TH_2(M)$ has no $2$-torsion and no $3$-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 | + | ** Even the computation of $\pi_0\SDiff(M)$ still leaves an unsolved extension problem in (\ref{eq:mcg}) above. |
</wikitex> | </wikitex> | ||
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== External links == | == External links == | ||
* The Wikipedia page on [[Wikipedia:5-manifold|1-connected 5-manifolds]] | * The Wikipedia page on [[Wikipedia:5-manifold|1-connected 5-manifolds]] | ||
− | <!-- --> | + | <!-- {{MediaWiki:Being refereed again}} --> |
[[Category:Manifolds]] | [[Category:Manifolds]] | ||
[[Category:Highly-connected manifolds]] | [[Category:Highly-connected manifolds]] |
Latest revision as of 22:07, 12 November 2016
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. |
The user responsible for this page is Diarmuid Crowley. No other user may edit this page at present. |
Contents |
1 Introduction
Tex syntax errorbe the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds
Tex syntax errorand let
Tex syntax errorbe the subset of diffeomorphism classes of spinable manifolds. The calculation of
Tex syntax errorwas 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
Tex syntax error.
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:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standard inclusion of .
In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of 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 , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex with only -cells and -cells and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
2.2 The general non-spin case
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 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.
Let be a pair with a surjective homomorphism and as above. We shall construct a non-spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection .
If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary .
In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-spin manifold as described above.
3 Invariants
Tex syntax error:
- be the second integral homology group of
Tex syntax error
, - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, , - , the smallest extended natural number such that for some . If
Tex syntax error
is spinable we set .
For example, the manifold has invariants , non-trivial and . The Wu-manifold, , has invariants , non-trivial and .
The above list is the minimal list of invariants required to give the classification of closed simply-connected -manifolds: see Theorem 4.2 below.
In addition we mention two further invariants ofTex syntax error:
- , the third Stiefel-Whitney class,
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on , the torsion subgroup of .
By [Milnor&Stasheff1974, Problem 8-A], , and so determines .
By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity where we regard as an element of . 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 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.
3.1 Linking forms
An abstract non-singular anti-symmetric linking form on a finite abelian group is a bi-linear function
such that for all if and only if and for all pairs and . For example, if denotes the cyclic group of order , we have the following linking forms specified by their linking matricies,
If is the sum of cyclic groups we shall write for the sum .
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
Tex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3.2 Values for constructions
The spin manifolds all have vanishing of course and so by Wall's classification of linking forms we see that the linking form of is the linking form .
As we mentioned above, the non-spin manifolds have given by projecting to and then applying :
If has height finite height then it follows from Wall's classification of linking forms that where and if has infinite height then .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets of .
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
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 and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism .
This theorem can re-phrased in categorical language as follows.
- Let be the groupoid with objects where is a finitely generated abelian group, is an anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
- Let be the groupoid with objects simply-connected closed smooth -manifolds embedded in some fixed for large and morphisms isotopy classes of diffeomorphisms.
- Consider the functor
Theorem 4.4 [Barden1965]. The functor 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 , [Barden1965, Theorem 2.3].
- , , , .
- For , is the spin manifold with constructed above.
- For let constructed above be the non-spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
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 -manifold embeds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
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 , and admits a unique spin structure which extends to we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
Every manifold 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 -manifoldTex syntax erroradmits a contact structure if and only if has an integral lift in . Hence
Tex syntax erroradmits a contact structure if and only if or ; equivalently
Tex syntax erroradmits a contact structure if and only if or where .
Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of prime to 3 was proven in [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, 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 for a general simply-connected 5-manifold in the literature.
- However if has no -torsion and no -torsion then was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
- Even the computation of 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 -connected -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 -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 -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 -connected -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 -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
Tex syntax errorand let
Tex syntax errorbe the subset of diffeomorphism classes of spinable manifolds. The calculation of
Tex syntax errorwas 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
Tex syntax error.
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:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standard inclusion of .
In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of 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 , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex with only -cells and -cells and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
2.2 The general non-spin case
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 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.
Let be a pair with a surjective homomorphism and as above. We shall construct a non-spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection .
If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary .
In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-spin manifold as described above.
3 Invariants
Tex syntax error:
- be the second integral homology group of
Tex syntax error
, - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, , - , the smallest extended natural number such that for some . If
Tex syntax error
is spinable we set .
For example, the manifold has invariants , non-trivial and . The Wu-manifold, , has invariants , non-trivial and .
The above list is the minimal list of invariants required to give the classification of closed simply-connected -manifolds: see Theorem 4.2 below.
In addition we mention two further invariants ofTex syntax error:
- , the third Stiefel-Whitney class,
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on , the torsion subgroup of .
By [Milnor&Stasheff1974, Problem 8-A], , and so determines .
By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity where we regard as an element of . 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 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.
3.1 Linking forms
An abstract non-singular anti-symmetric linking form on a finite abelian group is a bi-linear function
such that for all if and only if and for all pairs and . For example, if denotes the cyclic group of order , we have the following linking forms specified by their linking matricies,
If is the sum of cyclic groups we shall write for the sum .
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
Tex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3.2 Values for constructions
The spin manifolds all have vanishing of course and so by Wall's classification of linking forms we see that the linking form of is the linking form .
As we mentioned above, the non-spin manifolds have given by projecting to and then applying :
If has height finite height then it follows from Wall's classification of linking forms that where and if has infinite height then .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets of .
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
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 and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism .
This theorem can re-phrased in categorical language as follows.
- Let be the groupoid with objects where is a finitely generated abelian group, is an anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
- Let be the groupoid with objects simply-connected closed smooth -manifolds embedded in some fixed for large and morphisms isotopy classes of diffeomorphisms.
- Consider the functor
Theorem 4.4 [Barden1965]. The functor 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 , [Barden1965, Theorem 2.3].
- , , , .
- For , is the spin manifold with constructed above.
- For let constructed above be the non-spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
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 -manifold embeds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
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 , and admits a unique spin structure which extends to we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
Every manifold 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 -manifoldTex syntax erroradmits a contact structure if and only if has an integral lift in . Hence
Tex syntax erroradmits a contact structure if and only if or ; equivalently
Tex syntax erroradmits a contact structure if and only if or where .
Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of prime to 3 was proven in [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, 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 for a general simply-connected 5-manifold in the literature.
- However if has no -torsion and no -torsion then was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
- Even the computation of 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 -connected -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 -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 -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 -connected -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 -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
Tex syntax errorand let
Tex syntax errorbe the subset of diffeomorphism classes of spinable manifolds. The calculation of
Tex syntax errorwas 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
Tex syntax error.
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:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standard inclusion of .
In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of 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 , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex with only -cells and -cells and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
2.2 The general non-spin case
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 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.
Let be a pair with a surjective homomorphism and as above. We shall construct a non-spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection .
If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary .
In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-spin manifold as described above.
3 Invariants
Tex syntax error:
- be the second integral homology group of
Tex syntax error
, - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, , - , the smallest extended natural number such that for some . If
Tex syntax error
is spinable we set .
For example, the manifold has invariants , non-trivial and . The Wu-manifold, , has invariants , non-trivial and .
The above list is the minimal list of invariants required to give the classification of closed simply-connected -manifolds: see Theorem 4.2 below.
In addition we mention two further invariants ofTex syntax error:
- , the third Stiefel-Whitney class,
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on , the torsion subgroup of .
By [Milnor&Stasheff1974, Problem 8-A], , and so determines .
By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity where we regard as an element of . 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 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.
3.1 Linking forms
An abstract non-singular anti-symmetric linking form on a finite abelian group is a bi-linear function
such that for all if and only if and for all pairs and . For example, if denotes the cyclic group of order , we have the following linking forms specified by their linking matricies,
If is the sum of cyclic groups we shall write for the sum .
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
Tex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3.2 Values for constructions
The spin manifolds all have vanishing of course and so by Wall's classification of linking forms we see that the linking form of is the linking form .
As we mentioned above, the non-spin manifolds have given by projecting to and then applying :
If has height finite height then it follows from Wall's classification of linking forms that where and if has infinite height then .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets of .
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
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 and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism .
This theorem can re-phrased in categorical language as follows.
- Let be the groupoid with objects where is a finitely generated abelian group, is an anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
- Let be the groupoid with objects simply-connected closed smooth -manifolds embedded in some fixed for large and morphisms isotopy classes of diffeomorphisms.
- Consider the functor
Theorem 4.4 [Barden1965]. The functor 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 , [Barden1965, Theorem 2.3].
- , , , .
- For , is the spin manifold with constructed above.
- For let constructed above be the non-spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
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 -manifold embeds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
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 , and admits a unique spin structure which extends to we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
Every manifold 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 -manifoldTex syntax erroradmits a contact structure if and only if has an integral lift in . Hence
Tex syntax erroradmits a contact structure if and only if or ; equivalently
Tex syntax erroradmits a contact structure if and only if or where .
Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of prime to 3 was proven in [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, 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 for a general simply-connected 5-manifold in the literature.
- However if has no -torsion and no -torsion then was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
- Even the computation of 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 -connected -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 -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 -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 -connected -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 -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
Tex syntax errorand let
Tex syntax errorbe the subset of diffeomorphism classes of spinable manifolds. The calculation of
Tex syntax errorwas 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
Tex syntax error.
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:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standard inclusion of .
In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of 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 , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex with only -cells and -cells and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
2.2 The general non-spin case
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 4.3 entails that these choices do not affect the diffeomorphism type of the manifold constructed.
Let be a pair with a surjective homomorphism and as above. We shall construct a non-spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection .
If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary .
In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-spin manifold as described above.
3 Invariants
Tex syntax error:
- be the second integral homology group of
Tex syntax error
, - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, , - , the smallest extended natural number such that for some . If
Tex syntax error
is spinable we set .
For example, the manifold has invariants , non-trivial and . The Wu-manifold, , has invariants , non-trivial and .
The above list is the minimal list of invariants required to give the classification of closed simply-connected -manifolds: see Theorem 4.2 below.
In addition we mention two further invariants ofTex syntax error:
- , the third Stiefel-Whitney class,
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on , the torsion subgroup of .
By [Milnor&Stasheff1974, Problem 8-A], , and so determines .
By [Wall1962, Proposition 1 & 2], the linking form satisfies the identity where we regard as an element of . 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 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.
3.1 Linking forms
An abstract non-singular anti-symmetric linking form on a finite abelian group is a bi-linear function
such that for all if and only if and for all pairs and . For example, if denotes the cyclic group of order , we have the following linking forms specified by their linking matricies,
If is the sum of cyclic groups we shall write for the sum .
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
Tex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
3.2 Values for constructions
The spin manifolds all have vanishing of course and so by Wall's classification of linking forms we see that the linking form of is the linking form .
As we mentioned above, the non-spin manifolds have given by projecting to and then applying :
If has height finite height then it follows from Wall's classification of linking forms that where and if has infinite height then .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets of .
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
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 and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism .
This theorem can re-phrased in categorical language as follows.
- Let be the groupoid with objects where is a finitely generated abelian group, is an anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
- Let be the groupoid with objects simply-connected closed smooth -manifolds embedded in some fixed for large and morphisms isotopy classes of diffeomorphisms.
- Consider the functor
Theorem 4.4 [Barden1965]. The functor 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 , [Barden1965, Theorem 2.3].
- , , , .
- For , is the spin manifold with constructed above.
- For let constructed above be the non-spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
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 -manifold embeds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
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 , and admits a unique spin structure which extends to we see that every closed spin 5-manifold bounds a spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
Every manifold 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 -manifoldTex syntax erroradmits a contact structure if and only if has an integral lift in . Hence
Tex syntax erroradmits a contact structure if and only if or ; equivalently
Tex syntax erroradmits a contact structure if and only if or where .
Remark 5.2. The special case of this theorem for spin 5-manifolds with the order of prime to 3 was proven in [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying Theorem 4.3 above we obtain the following exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962a], , the group of homotopy -spheres. But by [Kervaire&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, 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 for a general simply-connected 5-manifold in the literature.
- However if has no -torsion and no -torsion then was computed in [Fang1993]. This computation agrees with a more recent conjectured answer: please see the discussion page.
- Even the computation of 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 -connected -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 -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 -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 -connected -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 -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