6-manifolds: 2-connected

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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
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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 $M$.
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Let $\mathcal{M}_6(0)$ 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 notation is used to be consistent with [[6-manifolds: 1-connected]]).
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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]]
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The classification $\mathcal{M}_6(0)$ 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]]
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$$ M \cong \sharp_r(S^3 \times S^3)$$
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$$ M \cong \#_r(S^3 \times S^3)$$
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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 $M$
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where by definition $\#_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$
$$ \chi(M) = 2 - 2r.$$
$$ \chi(M) = 2 - 2r.$$
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* For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
</wikitex>
</wikitex>
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The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
* $S^6$, the standard 6-sphere.
* $S^6$, the standard 6-sphere.
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* $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$.
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* $\#_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$.
</wikitex>
</wikitex>
== Invariants ==
== Invariants ==
<wikitex>;
<wikitex>;
−
Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:
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Suppose that $M$ is diffeomorphic to $\#_b(S^3 \times S^3)$ 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 $M$ is given by $b_3(M) = 2b$,
* 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 Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$,
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* 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 $\Zz^2$.
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* the [[Intersection form|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard [[Intersection form#Skew-symmetric bilinear forms|skew-symmetric hyperbolic]] form on $\Zz^2$.
</wikitex>
</wikitex>
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{{endthm}}
{{endthm}}
Hence if $\Nn$ denotes the natural numbers we obtain a bijection
Hence if $\Nn$ denotes the natural numbers we obtain a bijection
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$$ \mathcal{M}_6\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$
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$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$
</wikitex>
</wikitex>
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=== Topological 2-connected 6-manifolds ===
=== Topological 2-connected 6-manifolds ===
<wikitex>;
<wikitex>;
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Let $\mathcal{M}^{\Top}_6$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
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Let $\mathcal{M}^{\Top}_6(0)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
{{beginthm|Theorem}}
{{beginthm|Theorem}}
Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection
Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection
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$$ \mathcal{M}_6 \rightarrow \mathcal{M}^{\Top}_6.$$
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$$ \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$$
{{endthm}}
{{endthm}}
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=== Mapping class groups ===
=== Mapping class groups ===
<wikitex>;
<wikitex>;
−
...
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Let $\pi_0\Diff_+(M)$ denote the group of isotopy classes of diffeomorphisms $f \colon M \to M$ of a $2$-connected $6$-manifold $M$ and let $\Aut(M)$ denote the group of isomorphisms of $H_3(M)$ perserving the intersection form: $\Aut(M) \cong Sp_{2b}(\Zz)$ is the symplectic group when $M = \#_b(S^3 \times S^3)$. By \cite{Cerf1970} the forgetful map to the group of orientation preserving pseudo-isotopy classes of $M$ is an isomorphism. Applying Cerf's theorem Kreck proved in \cite{Kreck1979} that there are exact sequences
−
<wikitex>
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$$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast) $$
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$$ 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$$
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where by definition $\pi_0\SDiff(M)$ is the subgroup of isotopy classes induced the identity on $H_*(M)$ and $\Theta_7 \cong \pi_0(\Diff(D^6, \partial))$ is the group of [[Exotic spheres|homotopy $7$-spheres]].
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In particular $\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$.
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For more information about the extensions in $(\ast)$ above, see \cite{Krylov2003}, \cite{Johnson1983} and \cite{Crowley2009}.</wikitex>
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=== Diffeomorphism groups ===
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<wikitex>;
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Let $\mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6)$ denote group of diffeomorphisms of $\#_b S^3 \times S^3$ which are the identity inside a marked disc, and $B\mathcal{D}_b$ denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms $\mathcal{D}_b \to \mathcal{D}_{b+1}$, and so continuous maps $\mathcal{S} : B\mathcal{D}_b \to B\mathcal{D}_{b+1}$. The homology of these classifying spaces is approachable in a range of degrees, by the following theorem.
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{{beginthm|Theorem|{{cite|Galatius&Randal-Williams2012|Theorem 1.2}}}}
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The map
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$$\mathcal{S}_* : H_k(B\mathcal{D}_b) \longrightarrow H_k(B\mathcal{D}_{b+1})$$
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is an isomorphism for $b \geq 2k+4$.
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{{endthm}}
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Let $B\mathrm{Spin}(6)$ denote the classifying space of the group $\mathrm{Spin}(6)$, and $\gamma_6^{\mathrm{Spin}}$ denote its universal vector bundle. We write $MT\mathrm{Spin}(6)$ for the Thom spectrum of the virtual bundle $-\gamma_6^{\mathrm{Spin}}$. Pontrjagin--Thom theory provides a map
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$$\alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)$$
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to the basepoint component of the infinite loop space of the spectrum $MT\mathrm{Spin}(6)$, and these fit together under the maps $\mathcal{S}$ to give a map
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$$\alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).$$
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+
{{beginthm|Theorem|{{cite|Galatius&Randal-Williams2012a|Corollary 1.2}}}}
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The map $\alpha$ induces an isomorphism on (co)homology.
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{{endthm}}
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It is not difficult to calculate the rational cohomology of $\Omega^\infty_\bullet MT\mathrm{Spin}(6)$, and find that it is a polynomial algebra with generators in degrees $2,2,4,6,6,6,8,8,10,10,10,12,12,\ldots$, 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 $B\mathcal{D}_b$ in degrees $* \leq (b-4)/2$.
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</wikitex>
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== References ==
== References ==
{{#RefList:}}
{{#RefList:}}

Latest revision as of 17:54, 12 April 2012

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

[edit] 1 Introduction

Let 
Tex syntax error
 be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds 
Tex syntax error
 (the notation is used to be consistent with 6-manifolds: 1-connected). The classification 
Tex syntax error
 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 
Tex syntax error
 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 and in general 
Tex syntax error
 is determined by the formula for the Euler characteristic of 
Tex syntax error
 
Tex syntax error
For the more general case where 
Tex syntax error
, see 6-manifolds: 1-connected.

[edit] 2 Construction and examples

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

  • Tex syntax error
    , the standard 6-sphere.
  • \#_b(S^3 \times S^3), the 
    Tex syntax error
    -fold connected sum of 
    Tex syntax error
    .

[edit] 3 Invariants

Suppose that 
Tex syntax error
 is diffeomorphic to \#_b(S^3 \times S^3) then: 
  • Tex syntax error
    ,
  • the third Betti-number of 
    Tex syntax error
     is given by 
    Tex syntax error
    ,
  • the Euler characteristic of 
    Tex syntax error
     is given by 
    Tex syntax error
    ,
  • the intersection form of 
    Tex syntax error
     is isomorphic to the sum of b-copies of 
    Tex syntax error
    , the standard skew-symmetric hyperbolic form on 
    Tex syntax error
    .

[edit] 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 
Tex syntax error
.
Hence if 
Tex syntax error
 denotes the natural numbers we obtain a bijection 
Tex syntax error

[edit] 5 Further discussion

[edit] 5.1 Topological 2-connected 6-manifolds

Let \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 
Tex syntax error
 we have 
Tex syntax error
 and so 
Tex syntax error
 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

[edit] 5.2 Mapping class groups

Let \pi_0\Diff_+(M) denote the group of isotopy classes of diffeomorphisms f \colon M \to M of a 2-connected 6-manifold 
Tex syntax error
 and let \Aut(M) denote the group of isomorphisms of H_3(M) perserving the intersection form: \Aut(M) \cong Sp_{2b}(\Zz) is the symplectic group when M = \#_b(S^3 \times S^3). By [Cerf1970] the forgetful map to the group of orientation preserving pseudo-isotopy classes of 
Tex syntax error
 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) is the subgroup of isotopy classes induced the identity on H_*(M) and \Theta_7 \cong \pi_0(\Diff(D^6, \partial)) is the group of homotopy 7-spheres.

In particular \pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7.

For more information about the extensions in (\ast) above, see [Krylov2003], [Johnson1983] and [Crowley2009].

[edit] 5.3 Diffeomorphism groups

Let \mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6) denote group of diffeomorphisms of \#_b S^3 \times S^3 which are the identity inside a marked disc, and 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}, and so continuous maps \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.

Let B\mathrm{Spin}(6) denote the classifying space of the group \mathrm{Spin}(6), and \gamma_6^{\mathrm{Spin}} denote its universal vector bundle. We write MT\mathrm{Spin}(6) for the Thom spectrum of the virtual bundle -\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), and these fit together under the maps \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 induces an isomorphism on (co)homology.

It is not difficult to calculate the rational cohomology of \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, 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 in degrees * \leq (b-4)/2.

[edit] 6 References

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