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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=== 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. | + | 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) \ | + | $$ \mathcal{M}_6(0) \equiv\mathcal{M}^{\Top}_6(0).$$ |
{{endthm}} | {{endthm}} | ||
Line 53: | Line 53: | ||
=== 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. Applying Cerf's theorem Kreck proved in \cite{Kreck1979} that there are exact sequences |
− | <wikitex> | + | $$ 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast) $$ |
+ | $$ 0 \rightarrow \Theta_7 \rightarrow \pi_0\SDiff(M) \rightarrow H^3(M) \rightarrow 0$$ | ||
+ | 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 [[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> | ||
+ | |||
+ | === 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 |
1 Introduction
Tex syntax error(the notation is used to be consistent with 6-manifolds: 1-connected). The classification 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 erroris diffeomorphic to a connected-sum
Tex syntax error
For the more general case where , see 6-manifolds: 1-connected.
2 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
- , the standard 6-sphere.
- , the -fold connected sum of .
3 Invariants
Tex syntax erroris diffeomorphic to then:
- ,
- the third Betti-number of
Tex syntax error
is given by , - the Euler characteristic of
Tex syntax error
is given by , - the intersection form of
Tex syntax error
is isomorphic to the sum of b-copies of , the standard skew-symmetric hyperbolic form on .
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 .
Hence if denotes the natural numbers we obtain a bijection
5 Further discussion
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 manifoldTex syntax errorwe have and so
Tex syntax erroris smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.
5.2 Mapping class groups
Tex syntax errorand 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
Tex syntax erroris 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].
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 .
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