6-manifolds: 2-connected

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1 Introduction

Let $\mathcal{M}_6(0)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \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 \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\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 $\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

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

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$

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

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

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

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$$M$ we have $H^4(M; \Zz/2) \cong 0$$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

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

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

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