# 6-manifolds: 2-connected

## 1 Introduction


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

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

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$

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

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$$S^6$, the standard 6-sphere.
• $\#_b(S^3 \times S^3)$$\#_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:

• $\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

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

## 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

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

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

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

$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)$
$\displaystyle 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$

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

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

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

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

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

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

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$.