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]] | ||
− | $$ M \cong \ | + | $$ M \cong \#_r(S^3 \times S^3)$$ |
− | where by definition $\ | + | where by definition $\#_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$ |
$$ \chi(M) = 2 - 2r.$$ | $$ \chi(M) = 2 - 2r.$$ | ||
For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]]. | ||
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The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: | The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism: | ||
* $S^6$, the standard 6-sphere. | * $S^6$, the standard 6-sphere. | ||
− | * $\ | + | * $\#_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$. |
</wikitex> | </wikitex> | ||
== Invariants == | == Invariants == | ||
<wikitex>; | <wikitex>; | ||
− | Suppose that $M$ is diffeomorphic to $\ | + | Suppose that $M$ is diffeomorphic to $\#_b(S^3 \times S^3)$ then: |
* $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, | * $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$, | ||
* the third Betti-number of $M$ is given by $b_3(M) = 2b$, | * the third Betti-number of $M$ is given by $b_3(M) = 2b$, | ||
* the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$, | * the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$, | ||
− | * the [[Intersection | + | * the [[Intersection form|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard [[Intersection form#Skew-symmetric bilinear forms|skew-symmetric hyperbolic]] form on $\Zz^2$. |
</wikitex> | </wikitex> | ||
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{{beginthm|Theorem}} | {{beginthm|Theorem}} | ||
Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection | Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection | ||
− | $$ \mathcal{M}_6(0) \ | + | $$ \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$$ |
{{endthm}} | {{endthm}} | ||
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=== Mapping class groups === | === Mapping class groups === | ||
<wikitex>; | <wikitex>; | ||
− | 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. | + | 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 |
$$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast) $$ | $$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast) $$ | ||
− | $$ 0 \rightarrow \Theta_7 \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$$ | + | $$ 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$$ |
− | where by definition $\SDiff(M)$ is the subgroup of isotopy classes induced the identity on $H_*(M)$ | + | 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 $\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$. | In particular $\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7$. | ||
− | For more information about the | + | For more information about the extensions in $(\ast)$ above, see \cite{Krylov2003}, \cite{Johnson1983} and \cite{Crowley2009}.</wikitex> |
+ | |||
+ | === Diffeomorphism groups === | ||
+ | <wikitex>; | ||
+ | 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. | ||
+ | |||
+ | {{beginthm|Theorem|{{cite|Galatius&Randal-Williams2012|Theorem 1.2}}}} | ||
+ | The map | ||
+ | $$\mathcal{S}_* : H_k(B\mathcal{D}_b) \longrightarrow H_k(B\mathcal{D}_{b+1})$$ | ||
+ | is an isomorphism for $b \geq 2k+4$. | ||
+ | {{endthm}} | ||
+ | |||
+ | 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}}}} | ||
+ | The map $\alpha$ induces an isomorphism on (co)homology. | ||
+ | {{endthm}} | ||
+ | |||
+ | 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> | ||
+ | |||
== 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
Tex syntax errorbe the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds (the notation is used to be consistent with 6-manifolds: 1-connected). The classification
Tex syntax errorwas 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 is diffeomorphic to a connected-sum
where by definition and in general is determined by the formula for the Euler characteristic of
Tex syntax error
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. - , the -fold connected sum of
Tex syntax error
.
[edit] 3 Invariants
Suppose that is diffeomorphic to then:
-
Tex syntax error
, - the third Betti-number of is given by
Tex syntax error
, - the Euler characteristic of is given by
Tex syntax error
, - the intersection form of is isomorphic to the sum of b-copies of
Tex syntax error
, the standard skew-symmetric hyperbolic form on .
[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 byTex syntax error.
Tex syntax errordenotes the natural numbers we obtain a bijection
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[edit] 5 Further discussion
[edit] 5.1 Topological 2-connected 6-manifolds
Let 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
Proof.
For any such manifold we haveTex syntax errorand so is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.
[edit] 5.2 Mapping class groups
Let denote the group of isotopy classes of diffeomorphisms of a -connected -manifold and let denote the group of isomorphisms of perserving the intersection form: is the symplectic group when . By [Cerf1970] the forgetful map to the group of orientation preserving pseudo-isotopy classes of is an isomorphism. Applying Cerf's theorem Kreck proved in [Kreck1979] that there are exact sequences
where by definition is the subgroup of isotopy classes induced the identity on and is the group of homotopy -spheres.
In particular .
For more information about the extensions in above, see [Krylov2003], [Johnson1983] and [Crowley2009].
[edit] 5.3 Diffeomorphism groups
Let denote group of diffeomorphisms of which are the identity inside a marked disc, and denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms , and so continuous maps . The homology of these classifying spaces is approachable in a range of degrees, by the following theorem.
Let denote the classifying space of the group , and denote its universal vector bundle. We write for the Thom spectrum of the virtual bundle . Pontrjagin--Thom theory provides a map
to the basepoint component of the infinite loop space of the spectrum , and these fit together under the maps to give a map
Theorem 5.3 [Galatius&Randal-Williams2012a, Corollary 1.2]. The map induces an isomorphism on (co)homology.
It is not difficult to calculate the rational cohomology of , and find that it is a polynomial algebra with generators in degrees , 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 in degrees .
[edit] 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 for , (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 -connected almost-parallelizable -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 , 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