6-manifolds: 1-connected

1 Introduction

Let $\mathcal{M}_{6}(e)$$<wikitex>; == Introduction == Let $\mathcal{M}_{6}(e)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]], [[wikipedia:Differentiable_manifold|smooth]], [[wikipedia:Simply-connected|simply-connected]] 6-manifolds $M$. Similarly, let $\mathcal{M}^{\Top}_{6}(e)$ be the set of [[wikipedia:Homeomorphism|homeomorphism]] classes of closed, oriented [[wikipedia:Topological_manifold|topological manifolds]]. In this article we report on the calculation of $\mathcal{M}^{\Cat}_{6}(e)$ and $\mathcal{M}^{\Top}_{6}(e)$ begun by Smale, extended by Wall and Jupp and completed by Zhubr. We shall write $\mathcal{M}^{\Cat}_{6}(e)$ for either $\mathcal{M}^{}_{6}(e)$ or $\mathcal{M}^{\Top}_{6}(e)$. == Examples and constructions == We first present some familiar 6-manifolds. * $S^6$, the standard 6-sphere. * $\sharp_b S^3 \times S^3$, the $b$-fold connected sum of $S^3 \times S^3$. * $\sharp_r S^2 \times S^4$, the $r$-fold connected sum of $S^2 \times S^4$. * $\CP^3$, 3-dimensional complex projective space. * $S^2 \times_\gamma S^4$, the non-trivial linear 4-sphere bundle over $S^2$. * For each $\alpha \in \pi_3(\SO_3) \cong \Zz$ we have $S^4 \times_\alpha S^2$, the corresponding 2-sphere bundle over $S^4$. If we write 1 for a generator of $\pi_3(\SO_3)$ then $S^2 \times_1 S^4$ is diffeomorphic to $\CP^3$. Surgery on framed links. Let $\phi \co \sqcup_r S^3 \times D^3 \to S^6$ be a framed link. Then $M^6_\phi$, the outcome of surgery on $\phi$, is a simply connected Spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$. * ??? Complete intersections of some form. == Classification == === Smoothing theory === {{beginthm|Theorem 1}} Let $M$ be a simply-connected, topological 6-manifold. The [[wikipedia:Kirby–Siebenmann_class|Kirby-Siebenmann class]], $\KS(M) \in H^4(M; \Zz_2)$ is the sole obstruction to $M$ admitting a smooth structure. {{endthm}} {{beginthm|Theorem 2}} Every homeomorphism $f\co M \cong N$ of simply-connected, smooth $-manifolds is topologically isotopic to a diffeomorphism. Hence we have an injection $$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$ {{endthm}} === The second Stiefel-Whitney class === The second [[wikipedia:Stiefel–Whitney_class|Stiefel-Whitney class]] of $M$ is an element of $H^2(M; \Zz_2)$ which we regard as a homomorphism $w\co H_2(M) \rightarrow \Zz_2$. Let $\Hom({\mathcal Ab}, \Zz_2)$ be the set of isomorphism classes of pairs $(G, \omega)$ where $G$ is a finitely generated abelian group $w\co G \rightarrow \Zz_2$ is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to $\Zz_2$. The second Stiefel-Whitney classes defines a surjection $$ w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)$$ and we let $\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$ denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition $$ \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$ where $[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$. == 2-connected 6-manifolds == Smale showed that every smooth, 2-connected 6-manifold is diffeomorphic to $S^6$ or a connected sum $\sharp_r(S^3 \times S^3)$. Hence if $b_3(M)$ denotes the third Betti-number of $M$ and $\Nn$ denotes the natural numbers we obtain a bijection $$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$ Applying Theorems 1 and 2 we see that the same statement holds for $\mathcal{M}^{\Top}_6(0)$. === The splitting Theorem === {{beginthm|Theorem 3|(Wall)}} Let $M$ be a closed, smooth, simply-connected 6-manifold with $b_3(M) = 2r$. Then up to diffeomorphism, there is a unique maniofld $M_0$ with $b_3(M_0) = 0$ such that $M$ is diffeomorphic to $M_0 \sharp_r(S^3 \times S^3)$. {{endthm}} == 6-manifolds with torsion free second homology == In addition to $w\co H_2(M) \rightarrow \Zz_2$ and $b_3(M)$ we require the following invariants: * The first [[wikipedia:Pontrjagin_class|Pontrjagin class]] $p_1(M) \in H^4(M)$. * The Kirby-Siebenmann class $\KS(M) \in H^4(M; \Zz_2)$ * The cup product $\cup_3\co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$. These invariants satisfy the following relation $$W^3 = (p_1(M) + 24K) \cup W$$ for all $W \in H^2(M)$ which reduce to $w_2(M)$ mod $ and for all $K \in H^4(M)$ which reduce to $\KS(M)$ mod $. </nowiki> == References == * {{bibitem|Jupp1973}} * {{bibitem|Wall1966}} [[Category:Manifolds]] [[Category:Orientable]] {{MediaWiki:Stub}}\mathcal{M}_{6}(e)$ be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 6-manifolds $M$$M$. Similarly, let $\mathcal{M}^{\Top}_{6}(e)$$\mathcal{M}^{\Top}_{6}(e)$ be the set of homeomorphism classes of closed, oriented topological manifolds. In this article we report on the calculation of $\mathcal{M}^{\Cat}_{6}(e)$$\mathcal{M}^{\Cat}_{6}(e)$ and $\mathcal{M}^{\Top}_{6}(e)$$\mathcal{M}^{\Top}_{6}(e)$ begun by Smale, extended by Wall and Jupp and completed by Zhubr. We shall write $\mathcal{M}^{\Cat}_{6}(e)$$\mathcal{M}^{\Cat}_{6}(e)$ for either $\mathcal{M}^{}_{6}(e)$$\mathcal{M}^{}_{6}(e)$ or $\mathcal{M}^{\Top}_{6}(e)$$\mathcal{M}^{\Top}_{6}(e)$.

2 Examples and constructions

We first present some familiar 6-manifolds.

• $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

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  $S^3 \times S^3$.
• $\sharp_r S^2 \times S^4$$\sharp_r S^2 \times S^4$, the $r$$r$-fold connected sum of $S^2 \times S^4$$S^2 \times S^4$.
• $\CP^3$$\CP^3$, 3-dimensional complex projective space.
• $S^2 \times_\gamma S^4$$S^2 \times_\gamma S^4$, the non-trivial linear 4-sphere bundle over $S^2$$S^2$.
• For each $\alpha \in \pi_3(\SO_3) \cong \Zz$$\alpha \in \pi_3(\SO_3) \cong \Zz$ we have $S^4 \times_\alpha S^2$$S^4 \times_\alpha S^2$, the corresponding 2-sphere bundle over $S^4$$S^4$. If we write 1 for a generator of $\pi_3(\SO_3)$$\pi_3(\SO_3)$ then $S^2 \times_1 S^4$$S^2 \times_1 S^4$ is diffeomorphic to $\CP^3$$\CP^3$.

Surgery on framed links. Let $\phi \co \sqcup_r S^3 \times D^3 \to S^6$$\phi \co \sqcup_r S^3 \times D^3 \to S^6$ be a framed link. Then $M^6_\phi$$M^6_\phi$, the outcome of surgery on $\phi$$\phi$, is a simply connected Spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$$H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$$H_3(M_\phi) = 0$.

• ??? Complete intersections of some form.

3 Classification

3.1 Smoothing theory

Theorem 1 6.1. Let $M$$M$ be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, $\KS(M) \in H^4(M; \Zz_2)$$\KS(M) \in H^4(M; \Zz_2)$ is the sole obstruction to $M$$M$ admitting a smooth structure.

Theorem 2 6.2. Every homeomorphism $f\co M \cong N$$f\co M \cong N$ of simply-connected, smooth $6$$6$-manifolds is topologically isotopic to a diffeomorphism. Hence we have an injection

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

3.2 The second Stiefel-Whitney class

The second Stiefel-Whitney class of $M$$M$ is an element of $H^2(M; \Zz_2)$$H^2(M; \Zz_2)$ which we regard as a homomorphism $w\co H_2(M) \rightarrow \Zz_2$$w\co H_2(M) \rightarrow \Zz_2$. Let $\Hom({\mathcal Ab}, \Zz_2)$$\Hom({\mathcal Ab}, \Zz_2)$ be the set of isomorphism classes of pairs $(G, \omega)$$(G, \omega)$ where $G$$G$ is a finitely generated abelian group $w\co G \rightarrow \Zz_2$$w\co G \rightarrow \Zz_2$ is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to $\Zz_2$$\Zz_2$. The second Stiefel-Whitney classes defines a surjection

$\displaystyle w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)$ $$\displaystyle w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)$$

and we let $\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$$\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$ denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition

$\displaystyle \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$ $$\displaystyle \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$

where $[G, w]$$[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$$\Hom({\mathcal Ab}, \Zz_2)$.

4 2-connected 6-manifolds

Smale showed that every smooth, 2-connected 6-manifold is diffeomorphic to $S^6$$S^6$ or a connected sum $\sharp_r(S^3 \times S^3)$$\sharp_r(S^3 \times S^3)$. Hence if $b_3(M)$$b_3(M)$ denotes the third Betti-number of $M$$M$ and

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

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

Applying Theorems 1 and 2 we see that the same statement holds for $\mathcal{M}^{\Top}_6(0)$$\mathcal{M}^{\Top}_6(0)$.

4.1 The splitting Theorem

Theorem 3 8.1 (Wall). Let $M$$M$ be a closed, smooth, simply-connected 6-manifold with $b_3(M) = 2r$$b_3(M) = 2r$. Then up to diffeomorphism, there is a unique maniofld $M_0$$M_0$ with $b_3(M_0) = 0$$b_3(M_0) = 0$ such that $M$$M$ is diffeomorphic to $M_0 \sharp_r(S^3 \times S^3)$$M_0 \sharp_r(S^3 \times S^3)$.

5 6-manifolds with torsion free second homology

In addition to $w\co H_2(M) \rightarrow \Zz_2$$w\co H_2(M) \rightarrow \Zz_2$ and $b_3(M)$$b_3(M)$ we require the following invariants:

• The first Pontrjagin class $p_1(M) \in H^4(M)$$p_1(M) \in H^4(M)$.
• The Kirby-Siebenmann class $\KS(M) \in H^4(M; \Zz_2)$$\KS(M) \in H^4(M; \Zz_2)$
• The cup product $\cup_3\co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$$\cup_3\co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$.

These invariants satisfy the following relation

$\displaystyle W^3 = (p_1(M) + 24K) \cup W$ $$\displaystyle W^3 = (p_1(M) + 24K) \cup W$$

for all $W \in H^2(M)$$W \in H^2(M)$ which reduce to $w_2(M)$$w_2(M)$ mod $2$$2$ and for all $K \in H^4(M)$$K \in H^4(M)$ which reduce to $\KS(M)$$\KS(M)$ mod $2$$2$. </nowiki>

6 References

• [Jupp1973] P. E. Jupp, Classification of certain $6$$6$-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
• [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain $6$$6$-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601MediaWiki:Stub