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Introduction

Let $\mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_SNvHVl$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \DeclareMathOrd\GL{GL} % general linear group \DeclareMathOrd\SL{SL} % special linear group \DeclareMathOrd\SO{SO} % special orthogonal group \DeclareMathOrd\SU{SU} % special unitary group \DeclareMathOrd\Spin{Spin} % Spin group \DeclareMathOrd\RP{\Rr\mathrm P} % real projective space \DeclareMathOrd\CP{\Cc\mathrm P} % complex projective space \DeclareMathOrd\HP{\Hh\mathrm P} % quaternionic projective space \DeclareMathOrd\Top{\mathrm{Top}} % topological category \DeclareMathOrd\PL{\mathrm{PL}} % piecewise linear category \DeclareMathOrd\Cat{\mathrm{Cat}} % any category \DeclareMathOrd\KS{KS} % Kirby-Siebenmann class \DeclareMathOrd\Hud{Hud} % Hudson torus\newcommande{\ZZZ}{\mathbb{Z}}\mathcal{M}_6(0)$ be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_9k4byd$$M$.

The classification $\mathcal{M}_6(0)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_G4Acz5$$\mathcal{M}_6(0)$ was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The classification, as for oriented surfaces is strikingly simple: every 2-connected 6-manifold $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1BRi0X$$M$ is diffeomorphic to a connected-sum

$\displaystyle M \cong \sharp_r(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_u5XvTQ$ $$\displaystyle M \cong \sharp_r(S^3 \times S^3)$$

where by definition $\sharp_0(S^3 \times S^3) = S^6/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5zzGcK$$\sharp_0(S^3 \times S^3) = S^6$ and in general $r/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_yNViWD$$r$ is determined by the formula for the Euler characteristic of $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vPK15x$$M$

$\displaystyle \chi(M) = 2 - 2r./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_k4R2Es$ $$\displaystyle \chi(M) = 2 - 2r.$$

• For the more general case where $H_2(M) \neq 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_9hlJCp$$H_2(M) \neq 0$, see 6-manifolds: 1-connected.

1 Construction and examples

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

• $S^6/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_om5q3m$$S^6$, the standard 6-sphere.
• $\sharp_b(S^3 \times S^3)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_bBtHTk$$\sharp_b(S^3 \times S^3)$, the $b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_YtsHaj$$b$-fold connected sum of $S^3 \times S^3/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_hlhyRh$$S^3 \times S^3$.

2 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}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Wa4aYg$$\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
• the third Betti-number of $M$$M$ is given by $b_3(M) = 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Vnr5kh$$b_3(M) = 2b$,
• the Euler characteristic of $M$$M$ is given by $\chi(M) = 2 - 2b/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_cbT17h$$\chi(M) = 2 - 2b$,
• the intersection form of $M$$M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_zqRNkj$$H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_g5U2Wk$$\Zz^2$.

3 Classification

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

Theorem 5.1 [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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_BLDd2m$$\Nn$ denotes the natural numbers we obtain a bijection

$\displaystyle \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_SzHswp$ $$\displaystyle \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$

4 Further discussion

4.1 Topological 2-connected 6-manifolds

Let $\mathcal{M}^{\Top}_6(e)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_p079rs$$\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.

Theorem 6.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection

$\displaystyle \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e)./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_owYqSv$ $$\displaystyle \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$

Proof. For any such manifold $M$$M$ we have $H^4(M; \Zz/2) \cong 0/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1clzJz$$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 5.1 are diffeomorphic.

$\square$$\square$

4.2 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