6-manifolds: 2-connected

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1 Introduction

Let $\mathcal{M}_6$${{Stub}} == Introduction == <wikitex>; Let $\mathcal{M}_6$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 2-connected 6-manifolds $M$. The classification $\mathcal{M}_6$ was one of Smale's first applications of the [[Wikipedia:h-cobordism_theorem|h-cobordism]] theorem {{cite|Smale1962a|Corollary 1.3}}. The is a precise 6-dimensional analogue of the classification of [[Surface|orientable surfaces]]: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]] $$ M \cong \sharp_r(S^3 \times S^3)$$ where by definition $\sharp_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ $$ \chi(M) = 2 - 2r.$$ For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. </wikitex> == Construction and examples == <wikitex>; The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: * $S^6$, the standard 6-sphere. * $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$. </wikitex> == Invariants == <wikitex>; Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then: * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, * the third Betti-number of $M$ is given by $b_3(M) = 2b$, * the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$, * the [[Intersection forms|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$. </wikitex> == Classification == <wikitex>; Recall that the following theorem was stated in other words in the introduction: {{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification} The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$. {{endthm}} Hence if $\Nn$ denotes the natural numbers we obtain a bijection $$ \mathcal{M}_6\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ </wikitex> == Further discussion == === Topological 2-connected 6-manifolds === <wikitex>; Let $\mathcal{M}^{\Top}_6$ be the set of homeomorphism classes of topological 2-connected 6-manifolds. {{beginthm|Theorem}} Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection $$ \mathcal{M}_6 \rightarrow \mathcal{M}^{\Top}_6.$$ {{endthm}} {{beginproof}} For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see [[6-manifolds: 1-connected#Smoothing theory|6-manifolds: 1-connected]]). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem \ref{thm:classification} are diffeomorphic. {{endproof}} </wikitex> === Mapping class groups === <wikitex>; ... <wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Highly-connected manifolds]]\mathcal{M}_6$ be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds $M$$M$.

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

$\displaystyle M \cong \sharp_r(S^3 \times S^3)$ $$\displaystyle M \cong \sharp_r(S^3 \times S^3)$$

where by definition $\sharp_0(S^3 \times S^3) = S^6$$\sharp_0(S^3 \times S^3) = S^6$ and in general $r$$r$ is determined by the formula for the Euler characteristic of $M$$M$

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

For the more general case where $H_2(M) \neq 0$$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:

• $S^6$$S^6$, the standard 6-sphere.
• $\sharp_b(S^3 \times S^3)$$\sharp_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$$M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$$\sharp_b(S^3 \times S^3)$ then:

• $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$$\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
• the third Betti-number of $M$$M$ is given by $b_3(M) = 2b$$b_3(M) = 2b$,
• the Euler characteristic of $M$$M$ is given by $\chi(M) = 2 - 2b$$\chi(M) = 2 - 2b$,
• the intersection form of $M$$M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$$H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$$\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 $\Nn$$\Nn$ denotes the natural numbers we obtain a bijection

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

5 Further discussion

5.1 Topological 2-connected 6-manifolds

Let $\mathcal{M}^{\Top}_6$$\mathcal{M}^{\Top}_6$ 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

$\displaystyle \mathcal{M}_6 \rightarrow \mathcal{M}^{\Top}_6.$ $$\displaystyle \mathcal{M}_6 \rightarrow \mathcal{M}^{\Top}_6.$$

Proof. For any such manifold $M$$M$ we have $H^4(M; \Zz/2) \cong 0$$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

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References

• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103