6-manifolds: 2-connected

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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]]
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]]
$$ M \cong \sharp_r(S^3 \times S^3)$$
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$$ M \cong \#_r(S^3 \times S^3)$$
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 $M$
$$ \chi(M) = 2 - 2r.$$
$$ \chi(M) = 2 - 2r.$$
For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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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.
* $\sharp_b(S^3 \times S^3)$, 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 $\chi(M) = 2 - 2b$,
* 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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=== Topological 2-connected 6-manifolds ===
=== Topological 2-connected 6-manifolds ===
<wikitex>;
<wikitex>;
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
$$ \mathcal{M}_6(0) \rightarrow \mathcal{M}^{\Top}_6.$$
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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>
+
$$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast) $$
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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]].
+
+
In particular $\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$.
+
+
For more information about the extensions in $(\ast)$ above, see \cite{Krylov2003}, \cite{Johnson1983} and \cite{Crowley2009}.</wikitex>
+
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=== Diffeomorphism groups ===
+
<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
+
$$\alpha_b : B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6)$$
+
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
+
$$\alpha : \mathrm{hocolim}_{b \to \infty} B\mathcal{D}_b \longrightarrow \Omega^\infty_\bullet MT\mathrm{Spin}(6).$$
+
+
{{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 $* \leq (b-4)/2$.
+
</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 M (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 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 and in general r is determined by the formula for the Euler characteristic of M

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:

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

[edit] 3 Invariants

Suppose that M is diffeomorphic to \#_b(S^3 \times S^3) then:

  • Tex syntax error
    ,
  • the third Betti-number of M is given by
    Tex syntax error
    ,
  • the Euler characteristic of M is given by
    Tex syntax error
    ,
  • the intersection form of M is isomorphic to the sum of b-copies of
    Tex syntax error
    , the standard skew-symmetric hyperbolic form on \Zz^2.

[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 \Nn denotes the natural numbers we obtain a bijection

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

[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 M we have H^4(M; \Zz/2) \cong 0 and so 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

[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 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 [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 [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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