6-manifolds: 2-connected

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1 Introduction

Let

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${{Stub}} == Introduction == <wikitex>; Let $\mathcal{M}_6(0)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 2-connected 6-manifolds $M$ (the notation is used to be consistent with [[6-manifolds: 1-connected]]). The classification $\mathcal{M}_6(0)$ was one of Smale's first applications of the [[Wikipedia:h-cobordism_theorem|h-cobordism]] theorem {{cite|Smale1962a|Corollary 1.3}}. The is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] $$ M \cong \#_r(S^3 \times S^3)$$ where by definition $\#_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ $$ \chi(M) = 2 - 2r.$$ For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> == Construction and examples == <wikitex>; The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: * $S^6$, the standard 6-sphere. * $\#_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$. </wikitex> == Invariants == <wikitex>; Suppose that $M$ is diffeomorphic to $\#_b(S^3 \times S^3)$ then: * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, * the third Betti-number of $M$ is given by $b_3(M) = 2b$, * the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$, * the [[Intersection form|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard [[Intersection form#Skew-symmetric bilinear forms|skew-symmetric hyperbolic]] form on $\Zz^2$. </wikitex> == Classification == <wikitex>; Recall that the following theorem was stated in other words in the introduction: {{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification} The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$. {{endthm}} Hence if $\Nn$ denotes the natural numbers we obtain a bijection $$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ </wikitex> == Further discussion == === Topological 2-connected 6-manifolds === <wikitex>; Let $\mathcal{M}^{\Top}_6(0)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds. {{beginthm|Theorem}} Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection $$ \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$$ {{endthm}} {{beginproof}} For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see [[6-manifolds: 1-connected#Smoothing theory|6-manifolds: 1-connected]]). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem \ref{thm:classification} are diffeomorphic. {{endproof}} </wikitex> === Mapping class groups === <wikitex>; Let $\pi_0\Diff_+(M)$ denote the group of isotopy classes of diffeomorphisms $f \colon M \to M$ of a $-connected $-manifold $M$ and let $\Aut(M)$ denote the group of isomorphisms of $H_3(M)$ perserving the intersection form: $\Aut(M) \cong Sp_{2b}(\Zz)$ is the symplectic group when $M = \#_b(S^3 \times S^3)$. By \cite{Cerf1970} the forgetful map to the group of orientation preserving pseudo-isotopy classes of $M$ is an isomorphism. Applying Cerf's theorem Kreck proved in \cite{Kreck1979} that there are exact sequences $$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast) $$ $$ 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$$ where by definition $\pi_0\SDiff(M)$ is the subgroup of isotopy classes induced the identity on $H_*(M)$ and $\Theta_7 \cong \pi_0(\Diff(D^6, \partial))$ is the group of [[Exotic spheres|homotopy $-spheres]]. In particular $\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$. For more information about the extensions in $(\ast)$ above, see \cite{Krylov2003}, \cite{Johnson1983} and \cite{Crowley2009}.</wikitex> === Diffeomorphism groups === <wikitex>; Let $\mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6)$ denote group of diffeomorphisms of $\#_b S^3 \times S^3$ which are the identity inside a marked disc, and $B\mathcal{D}_b$ denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms $\mathcal{D}_b \to \mathcal{D}_{b+1}$, and so continuous maps $\mathcal{S} : B\mathcal{D}_b \to B\mathcal{D}_{b+1}$. The homology of these classifying spaces is approachable in a range of degrees, by the following theorem. {{beginthm|Theorem|{{cite|Galatius&Randal-Williams2012|Theorem 1.2}}}} The map $$\mathcal{S}_* : H_k(B\mathcal{D}_b) \longrightarrow H_k(B\mathcal{D}_{b+1})$$ is an isomorphism for $b \geq 2k+4$. {{endthm}} Let $B\mathrm{Spin}(6)$ denote the classifying space of the group $\mathrm{Spin}(6)$, and $\gamma_6^{\mathrm{Spin}}$ denote its universal vector bundle. We write $MT\mathrm{Spin}(6)$ for the Thom spectrum of the virtual bundle $-\gamma_6^{\mathrm{Spin}}$. Pontrjagin--Thom theory provides a map $$\alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)$$ to the basepoint component of the infinite loop space of the spectrum $MT\mathrm{Spin}(6)$, and these fit together under the maps $\mathcal{S}$ to give a map $$\alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).$$ {{beginthm|Theorem|{{cite|Galatius&Randal-Williams2012a|Corollary 1.2}}}} The map $\alpha$ induces an isomorphism on (co)homology. {{endthm}} It is not difficult to calculate the rational cohomology of $\Omega^\infty_\bullet MT\mathrm{Spin}(6)$, and find that it is a polynomial algebra with generators in degrees ,2,4,6,6,6,8,8,10,10,10,12,12,\ldots$, which can be given an explicit description in terms of generalised Miller-Morita-Mumford classes. By the stability theorem, this calculates the rational cohomology of $B\mathcal{D}_b$ in degrees $* \leq (b-4)/2$. </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Highly-connected manifolds]]\mathcal{M}_6(0)$ be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds $M$$M$ (the notation is used to be consistent with 6-manifolds: 1-connected). The classification

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$\mathcal{M}_6(0)$ was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The is a precise 6-dimensional analogue of the classification of orientable surfaces: every 2-connected 6-manifold $M$$M$ is diffeomorphic to a connected-sum

$\displaystyle M \cong \#_r(S^3 \times S^3)$ $$\displaystyle M \cong \#_r(S^3 \times S^3)$$

where by definition $\#_0(S^3 \times S^3) = S^6$$\#_0(S^3 \times S^3) = S^6$ and in general $r$$r$ is determined by the formula for the Euler characteristic of $M$$M$

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$$\displaystyle \chi(M) = 2 - 2r.$$

For the more general case where

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$H_2(M) \neq 0$, see 6-manifolds: 1-connected.

2 Construction and examples

The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:

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  $S^6$, the standard 6-sphere.
• $\#_b(S^3 \times S^3)$$\#_b(S^3 \times S^3)$, the $b$$b$-fold connected sum of $S^3 \times S^3$$S^3 \times S^3$.

3 Invariants

Suppose that $M$$M$ is diffeomorphic to $\#_b(S^3 \times S^3)$$\#_b(S^3 \times S^3)$ then:

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  $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
• the third Betti-number of $M$$M$ is given by

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  $b_3(M) = 2b$,
• the Euler characteristic of $M$$M$ is given by

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  $\chi(M) = 2 - 2b$,
• the intersection form of $M$$M$ is isomorphic to the sum of b-copies of

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  $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$$\Zz^2$.

4 Classification

Recall that the following theorem was stated in other words in the introduction:

Theorem 4.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$$S^3 \times S^3$.

Hence if

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$\Nn$ denotes the natural numbers we obtain a bijection

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$$\displaystyle \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$

5 Further discussion

5.1 Topological 2-connected 6-manifolds

Let $\mathcal{M}^{\Top}_6(0)$$\mathcal{M}^{\Top}_6(0)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.

Theorem 5.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection

$\displaystyle \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$ $$\displaystyle \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$$

Proof.

For any such manifold $M$$M$ we have

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$H^4(M; \Zz/2) \cong 0$ and so $M$$M$ is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.

$\square$$\square$

5.2 Mapping class groups

Let $\pi_0\Diff_+(M)$$\pi_0\Diff_+(M)$ denote the group of isotopy classes of diffeomorphisms $f \colon M \to M$$f \colon M \to M$ of a $2$$2$-connected $6$$6$-manifold $M$$M$ and let $\Aut(M)$$\Aut(M)$ denote the group of isomorphisms of $H_3(M)$$H_3(M)$ perserving the intersection form: $\Aut(M) \cong Sp_{2b}(\Zz)$$\Aut(M) \cong Sp_{2b}(\Zz)$ is the symplectic group when $M = \#_b(S^3 \times S^3)$$M = \#_b(S^3 \times S^3)$. By [Cerf1970] the forgetful map to the group of orientation preserving pseudo-isotopy classes of $M$$M$ is an isomorphism. Applying Cerf's theorem Kreck proved in [Kreck1979] that there are exact sequences

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

$\displaystyle 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$ $$\displaystyle 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$$

where by definition $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ is the subgroup of isotopy classes induced the identity on $H_*(M)$$H_*(M)$ and $\Theta_7 \cong \pi_0(\Diff(D^6, \partial))$$\Theta_7 \cong \pi_0(\Diff(D^6, \partial))$ is the group of homotopy $7$$7$-spheres.

In particular $\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$$\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$.

For more information about the extensions in $(\ast)$$(\ast)$ above, see [Krylov2003], [Johnson1983] and [Crowley2009].

5.3 Diffeomorphism groups

Let $\mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6)$$\mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6)$ denote group of diffeomorphisms of $\#_b S^3 \times S^3$$\#_b S^3 \times S^3$ which are the identity inside a marked disc, and $B\mathcal{D}_b$$B\mathcal{D}_b$ denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms $\mathcal{D}_b \to \mathcal{D}_{b+1}$$\mathcal{D}_b \to \mathcal{D}_{b+1}$, and so continuous maps $\mathcal{S} : B\mathcal{D}_b \to B\mathcal{D}_{b+1}$$\mathcal{S} : B\mathcal{D}_b \to B\mathcal{D}_{b+1}$. The homology of these classifying spaces is approachable in a range of degrees, by the following theorem.

Theorem 5.2 [Galatius&Randal-Williams2012, Theorem 1.2]. The map

$\displaystyle \mathcal{S}_* : H_k(B\mathcal{D}_b) \longrightarrow H_k(B\mathcal{D}_{b+1})$ $$\displaystyle \mathcal{S}_* : H_k(B\mathcal{D}_b) \longrightarrow H_k(B\mathcal{D}_{b+1})$$

is an isomorphism for $b \geq 2k+4$$b \geq 2k+4$.

Let $B\mathrm{Spin}(6)$$B\mathrm{Spin}(6)$ denote the classifying space of the group $\mathrm{Spin}(6)$$\mathrm{Spin}(6)$, and $\gamma_6^{\mathrm{Spin}}$$\gamma_6^{\mathrm{Spin}}$ denote its universal vector bundle. We write $MT\mathrm{Spin}(6)$$MT\mathrm{Spin}(6)$ for the Thom spectrum of the virtual bundle $-\gamma_6^{\mathrm{Spin}}$$-\gamma_6^{\mathrm{Spin}}$. Pontrjagin--Thom theory provides a map

$\displaystyle \alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)$ $$\displaystyle \alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)$$

to the basepoint component of the infinite loop space of the spectrum $MT\mathrm{Spin}(6)$$MT\mathrm{Spin}(6)$, and these fit together under the maps $\mathcal{S}$$\mathcal{S}$ to give a map

$\displaystyle \alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).$ $$\displaystyle \alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).$$

Theorem 5.3 [Galatius&Randal-Williams2012a, Corollary 1.2]. The map $\alpha$$\alpha$ induces an isomorphism on (co)homology.

It is not difficult to calculate the rational cohomology of $\Omega^\infty_\bullet MT\mathrm{Spin}(6)$$\Omega^\infty_\bullet MT\mathrm{Spin}(6)$, and find that it is a polynomial algebra with generators in degrees $2,2,4,6,6,6,8,8,10,10,10,12,12,\ldots$$2,2,4,6,6,6,8,8,10,10,10,12,12,\ldots$, which can be given an explicit description in terms of generalised Miller-Morita-Mumford classes. By the stability theorem, this calculates the rational cohomology of $B\mathcal{D}_b$$B\mathcal{D}_b$ in degrees $* \leq (b-4)/2$$* \leq (b-4)/2$.

6 References

• [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
• [Crowley2009] D. Crowley, On the mapping class groups of $\#_r(S^p \times S^p)$$\#_r(S^p \times S^p)$ for $p = 3, 7$$p = 3, 7$, (2009). Available at the arXiv:0905.0423.
• [Galatius&Randal-Williams2012] S. Galatius and O. Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, (2012). Available at the arXiv:1203.6830.
• [Galatius&Randal-Williams2012a] S. Galatius and O. Randal-Williams, Stable moduli spaces of high dimensional manifolds, (2012). Available at the arXiv:1201.3527.
• [Johnson1983] D. Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981), Amer. Math. Soc. (1983), 165–179. MR718141 (85d:57009) Zbl 0553.57002
• [Kreck1979] M. Kreck, Isotopy classes of diffeomorphisms of $(k-1)$$(k-1)$-connected almost-parallelizable $2k$$2k$-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Springer (1979), 643–663. MR561244 (81i:57029) Zbl 0421.57009
• [Krylov2003] N. A. Krylov, On the Jacobi group and the mapping class group of $S^3\times S^3$$S^3\times S^3$, Trans. Amer. Math. Soc. 355 (2003), no.1, 99–117 (electronic). MR1928079 (2003i:57039) Zbl 1015.57020
• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103