6-manifolds: 1-connected

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1 Introduction

Let $\mathcal{M}_{6}$${{Stub}} == Introduction == <wikitex>; Let $\mathcal{M}_{6} $ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]] [[wikipedia:Simply-connected|simply-connected]] 6-manifolds $M$. Similarly, let $\mathcal{M}^{PL}_{6}$ be the set of PL-homeomorphism classes of closed simply-connected PL 6-manifolds and let $\mathcal{M}^{TOP}_{6}$ be the set of [[wikipedia:Homeomorphism|homeomorphism]] classes of closed simply connected [[wikipedia:Topological_manifold|topological]] $-manifolds. In this article we report on the calculation of $\mathcal{M}^{}_{6}$, $\mathcal{M}^{PL}_{6}$ and $\mathcal{M}^{TOP}_{6}$ begun by {{cite|Smale1962}}, extended in {{cite|Wall1966}} and {{cite|Jupp1973}} and finally completed in {{cite|Zhubr2000}}. We shall write $\mathcal{M}^{\Cat}_{6}$ for either $\mathcal{M}^{}_{6}$, $\mathcal{M}^{PL}_{6}$ or $\mathcal{M}^{\Top}_{6}$. Our first theorem is a consequence of smoothing theory for $-manifolds. {{beginthm|Theorem}} The forgetful map from smooth to PL manifolds defines a bijection $\mathcal{M}_{6} \equiv \mathcal{M}^{PL}_{6}.$ {{endthm}} {{beginproof}} To be filled in. {{endproof}} An excellent summary for the case where $H_2(M)$ is torsion free may be found in {{cite|Okonek&Van de Ven1995|Section 1}}. For the case where $H_2(M) = 0$, see [[6-manifolds: 2-connected]]. </wikitex> == Examples and constructions == <wikitex>; We first present some familiar 6-manifolds. * $S^6$, the 6-sphere. * $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$. * $\sharp_r(S^2 \times S^4)$, the $r$-fold connected sum of $S^2 \times S^4$. * $\CP^3$, 3-dimensional complex projective space. * $S^4 \tilde \times_\gamma S^2$, the non-trivial linear 4-sphere bundle over $S^2$. * For each $\alpha \in \pi_3(\SO_3) \cong \Zz$ we have $S^2 \tilde \times_\alpha S^4$, the corresponding 2-sphere bundle over $S^4$. If we write 1 for a generator of $\pi_3(\SO_3)$ then $S^2 \tilde \times_1 S^4$ is diffeomorphic to $\CP^3$. * The smooth manifold underlying any complex manifold of dimension 3 is a 1-connected 6-manifold: ** in particular, every [[Complete intersections|complete intersection]] of complex dimension 3 is a 1-connected 6-manifold. * Let $\phi \co \sqcup_{i=1}^r (D^3 \times S^3) \to S^6$ be an $r$-component framed link and let denote by $M^6_\phi$ the outcome of surgery on $\phi$. Then $M_\phi$ is a simply connected spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$. </wikitex> == Invariants == <wikitex>; The following gives a list of the key invariants needed to classify 1-connected 6-manifolds $M$: * The 3rd Betti-number, $b_3(M)$ which is even since the intersection for of $M$ is skew-symmetric. * The second [[wikipedia:Stiefel–Whitney_class|Stiefel-Whitney class]] of $M$ is an element of $H^2(M; \Zz_2)$ which we regard as a homomorphism $w\co H_2(M) \rightarrow \Zz_2$. * The first [[wikipedia:Pontrjagin_class|Pontrjagin class]] $p_1(M) \in H^4(M)$. * The [[wikipedia:Kirby-Siebenmann class|Kirby-Siebenmann class]] $\KS(M) \in H^4(M; \Zz_2)$. * The cup product $F_M \co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$. These invariants satisfy the following relation $$W^3 = (p_1(M) + 24K) \cup W$$ for all $W \in H^2(M)$ which reduce to $w_2(M)$ mod $ and for all $K \in H^4(M)$ which reduce to $\KS(M)$ mod $. As {{cite|Okonek&Van de Ven1995|p. 300}} remark, in the smooth case this follows from the integrality of the $\hat A$-genus but in the topological case requires further arguments carried out in {{cite|Jupp1973}}. Note that if $b_3(M) = 2b$ then the [[Intersection form|intersection form]] of $M$ is isomorphic to $b$ copies of $H_{-1}(\Zz)$, the skew-symmetric hyperbolic form on $\Zz^2$. </wikitex> == Classification == In this section we organise the classification results for simply-connected 6-manifolds. === Notation === <wikitex>; Let $\Hom({\mathcal Ab}, \Zz_2)$ be the set of isomorphism classes of pairs $(G, \omega)$ where $G$ is a finitely generated abelian group $w\co G \rightarrow \Zz_2$ is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to $\Zz_2$. The second Stiefel-Whitney classes defines a surjection $$ w_2\co\mathcal{M}_{6}^{\Cat} \longrightarrow \Hom({\mathcal Ab}, \Zz_2), \quad \quad [M] \longmapsto w_2 : H_2(M) \to \Zz/2$$ and we let $\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$ denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition $$ \mathcal{M}^{\Cat}_{6} = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$ where $[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$. </wikitex> === Splitting Theorem === <wikitex>; The following theorem of Wall reduces the classification of smooth simply connected 6-manfiolds to the [[6-manifolds: 2-connected|2-connected case]] and the case where $H_3(M)$ is torsion free. {{beginthm|Theorem 3|{{cite|Wall1966|Theorem 1}}}} Let $M$ be a closed, smooth, simply-connected 6-manifold with $b_3(M) = 2b$. Then up to diffeomorphism, there is a unique maniofld $M_0$ with $b_3(M_0) = 0$ such that $M$ is diffeomorphic to $M_0 \sharp_r(S^3 \times S^3)$. {{endthm}} </wikitex> === Smoothing theory === <wikitex>; Let $M$ be a topological 6-manifold and recall the Kirby-Siebenmann invariant $\KS(M) \in H^4(M; \Zz/2)$ defined in \cite{Kirby&Siebenmann1977}}: from the far reaching results of this book we have the following {{beginthm|Theorem}} \label{thm:smoothing} Let $M$ be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, $\KS(M) \in H^4(M; \Zz_2)$ is the sole obstruction to $M$ admitting a smooth structure. {{endthm}} </wikitex> === The case H₂ is torsion free === <wikitex>; The paper {{cite|Zhubr2000}} contains a complete classification of all 1-connected 6-manifolds. However, the classification is rather complex. We state here only the classification in the case where $H_2(M)$ is torsion free. Recall that the following system of invariants $(b_3(M)/2, H^2(M), F_M, w_2(M), p_1(M), \KS(M))$. {{beginthm|Theorem|{{cite|Jupp1973}}}}\label{thm:classification} Let $M_0$ and $M_1$ be 1-connected 6-manifolds with $H_2(M)$ torsion free. Suppose that $A : H^2(M_0) \cong H^2(M_1)$ is an isomorphism of abelian groups such that * $A^*F_{M_1} = F_{M_2}$, * $A^*w_2(M_1) = w_2(M_0)$, * $A^*p_1(M_0) = p_1(M_1)$ and * $A^*\KS(M_0) = A^*\KS(M_1)$, then there is a homeomorphism $f : M_0 \cong M_1$ inducing $A$ on $H^2$. If, in addition, $\KS(M_0) = 0 = \KS(M_1)$, then $f$ may be chosen to be a diffeomorphism and $M_0 \cong M_1$ admits a unique smooth structure. {{endthm}} </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} [[Category:Manifolds]]\mathcal{M}_{6}$ be the set of diffeomorphism classes of closed smooth oriented simply-connected 6-manifolds $M$$M$.

Similarly, let $\mathcal{M}^{PL}_{6}$$\mathcal{M}^{PL}_{6}$ be the set of PL-homeomorphism classes of closed simply-connected PL 6-manifolds and let $\mathcal{M}^{TOP}_{6}$$\mathcal{M}^{TOP}_{6}$ be the set of homeomorphism classes of closed simply connected topological $6$$6$-manifolds.

In this article we report on the calculation of $\mathcal{M}^{}_{6}$$\mathcal{M}^{}_{6}$, $\mathcal{M}^{PL}_{6}$$\mathcal{M}^{PL}_{6}$ and $\mathcal{M}^{TOP}_{6}$$\mathcal{M}^{TOP}_{6}$ begun by [Smale1962], extended in [Wall1966] and [Jupp1973] and finally completed in [Zhubr2000]. We shall write $\mathcal{M}^{\Cat}_{6}$$\mathcal{M}^{\Cat}_{6}$ for either $\mathcal{M}^{}_{6}$$\mathcal{M}^{}_{6}$, $\mathcal{M}^{PL}_{6}$$\mathcal{M}^{PL}_{6}$ or $\mathcal{M}^{\Top}_{6}$$\mathcal{M}^{\Top}_{6}$.

Our first theorem is a consequence of smoothing theory for $6$$6$-manifolds.

Theorem 1.1. The forgetful map from smooth to PL manifolds defines a bijection $\mathcal{M}_{6} \equiv \mathcal{M}^{PL}_{6}.$$\mathcal{M}_{6} \equiv \mathcal{M}^{PL}_{6}.$

Proof. To be filled in.

$\square$$\square$

An excellent summary for the case where $H_2(M)$$H_2(M)$ is torsion free may be found in [Okonek&Van de Ven1995, Section 1].

For the case where $H_2(M) = 0$$H_2(M) = 0$, see 6-manifolds: 2-connected.

2 Examples and constructions

We first present some familiar 6-manifolds.

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  $S^6$, the 6-sphere.

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  $\sharp_b(S^3 \times S^3)$, the

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  $b$-fold connected sum of

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  $S^3 \times S^3$.
• $\sharp_r(S^2 \times S^4)$$\sharp_r(S^2 \times S^4)$, the

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  $r$-fold connected sum of $S^2 \times S^4$$S^2 \times S^4$.
• $\CP^3$$\CP^3$, 3-dimensional complex projective space.
• $S^4 \tilde \times_\gamma S^2$$S^4 \tilde \times_\gamma S^2$, the non-trivial linear 4-sphere bundle over $S^2$$S^2$.
• For each $\alpha \in \pi_3(\SO_3) \cong \Zz$$\alpha \in \pi_3(\SO_3) \cong \Zz$ we have $S^2 \tilde \times_\alpha S^4$$S^2 \tilde \times_\alpha S^4$, the corresponding 2-sphere bundle over $S^4$$S^4$. If we write 1 for a generator of $\pi_3(\SO_3)$$\pi_3(\SO_3)$ then $S^2 \tilde \times_1 S^4$$S^2 \tilde \times_1 S^4$ is diffeomorphic to $\CP^3$$\CP^3$.
• The smooth manifold underlying any complex manifold of dimension 3 is a 1-connected 6-manifold:
  • in particular, every complete intersection of complex dimension 3 is a 1-connected 6-manifold.
• Let $\phi \co \sqcup_{i=1}^r (D^3 \times S^3) \to S^6$$\phi \co \sqcup_{i=1}^r (D^3 \times S^3) \to S^6$ be an

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  $r$-component framed link and let denote by $M^6_\phi$$M^6_\phi$ the outcome of surgery on $\phi$$\phi$. Then $M_\phi$$M_\phi$ is a simply connected spinable 6-manifold with $H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$$H_2(M_\phi) \cong H_4(M_\phi) \cong \Zz^r$ and $H_3(M_\phi) = 0$$H_3(M_\phi) = 0$.

3 Invariants

The following gives a list of the key invariants needed to classify 1-connected 6-manifolds $M$$M$:

• The 3rd Betti-number, $b_3(M)$$b_3(M)$ which is even since the intersection for of $M$$M$ is skew-symmetric.
• The second Stiefel-Whitney class of $M$$M$ is an element of $H^2(M; \Zz_2)$$H^2(M; \Zz_2)$ which we regard as a homomorphism $w\co H_2(M) \rightarrow \Zz_2$$w\co H_2(M) \rightarrow \Zz_2$.
• The first Pontrjagin class $p_1(M) \in H^4(M)$$p_1(M) \in H^4(M)$.
• The Kirby-Siebenmann class $\KS(M) \in H^4(M; \Zz_2)$$\KS(M) \in H^4(M; \Zz_2)$.
• The cup product $F_M \co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$$F_M \co H^2(M) \otimes H^2(M) \otimes H^2(M) \rightarrow H^6(M) = \Zz$.

These invariants satisfy the following relation

$\displaystyle W^3 = (p_1(M) + 24K) \cup W$ $$\displaystyle W^3 = (p_1(M) + 24K) \cup W$$

for all $W \in H^2(M)$$W \in H^2(M)$ which reduce to $w_2(M)$$w_2(M)$ mod $2$$2$ and for all $K \in H^4(M)$$K \in H^4(M)$ which reduce to $\KS(M)$$\KS(M)$ mod $2$$2$. As [Okonek&Van de Ven1995, p. 300] remark, in the smooth case this follows from the integrality of the $\hat A$$\hat A$-genus but in the topological case requires further arguments carried out in [Jupp1973].

Note that if

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$b_3(M) = 2b$ then the intersection form of $M$$M$ is isomorphic to

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$b$ copies of $H_{-1}(\Zz)$$H_{-1}(\Zz)$, the skew-symmetric hyperbolic form on

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$\Zz^2$.

4 Classification

In this section we organise the classification results for simply-connected 6-manifolds.

4.1 Notation

Let $\Hom({\mathcal Ab}, \Zz_2)$$\Hom({\mathcal Ab}, \Zz_2)$ be the set of isomorphism classes of pairs $(G, \omega)$$(G, \omega)$ where $G$$G$ is a finitely generated abelian group $w\co G \rightarrow \Zz_2$$w\co G \rightarrow \Zz_2$ is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to $\Zz_2$$\Zz_2$. The second Stiefel-Whitney classes defines a surjection

$\displaystyle w_2\co\mathcal{M}_{6}^{\Cat} \longrightarrow \Hom({\mathcal Ab}, \Zz_2), \quad \quad [M] \longmapsto w_2 : H_2(M) \to \Zz/2$ $$\displaystyle w_2\co\mathcal{M}_{6}^{\Cat} \longrightarrow \Hom({\mathcal Ab}, \Zz_2), \quad \quad [M] \longmapsto w_2 : H_2(M) \to \Zz/2$$

and we let $\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$$\mathcal{M}^{\Cat}_6(G, w) = w^{-1}([G, w])$ denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition

$\displaystyle \mathcal{M}^{\Cat}_{6} = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$ $$\displaystyle \mathcal{M}^{\Cat}_{6} = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)$$

where $[G, w]$$[G, w]$ ranges over all of $\Hom({\mathcal Ab}, \Zz_2)$$\Hom({\mathcal Ab}, \Zz_2)$.

4.2 Splitting Theorem

The following theorem of Wall reduces the classification of smooth simply connected 6-manfiolds to the 2-connected case and the case where $H_3(M)$$H_3(M)$ is torsion free.

Theorem 3 4.1 [Wall1966, Theorem 1].

Let $M$$M$ be a closed, smooth, simply-connected 6-manifold with

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$b_3(M) = 2b$. Then up to diffeomorphism, there is a unique maniofld $M_0$$M_0$ with $b_3(M_0) = 0$$b_3(M_0) = 0$ such that $M$$M$ is diffeomorphic to $M_0 \sharp_r(S^3 \times S^3)$$M_0 \sharp_r(S^3 \times S^3)$.

4.3 Smoothing theory

Let $M$$M$ be a topological 6-manifold and recall the Kirby-Siebenmann invariant $\KS(M) \in H^4(M; \Zz/2)$$\KS(M) \in H^4(M; \Zz/2)$ defined in [Kirby&Siebenmann1977]}: from the far reaching results of this book we have the following

Theorem 4.2. Let $M$$M$ be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, $\KS(M) \in H^4(M; \Zz_2)$$\KS(M) \in H^4(M; \Zz_2)$ is the sole obstruction to $M$$M$ admitting a smooth structure.

4.4 The case H₂ is torsion free

The paper [Zhubr2000] contains a complete classification of all 1-connected 6-manifolds. However, the classification is rather complex. We state here only the classification in the case where $H_2(M)$$H_2(M)$ is torsion free.

Recall that the following system of invariants $(b_3(M)/2, H^2(M), F_M, w_2(M), p_1(M), \KS(M))$$(b_3(M)/2, H^2(M), F_M, w_2(M), p_1(M), \KS(M))$.

Theorem 4.3 [Jupp1973]. Let $M_0$$M_0$ and $M_1$$M_1$ be 1-connected 6-manifolds with $H_2(M)$$H_2(M)$ torsion free. Suppose that $A : H^2(M_0) \cong H^2(M_1)$$A : H^2(M_0) \cong H^2(M_1)$ is an isomorphism of abelian groups such that

• $A^*F_{M_1} = F_{M_2}$$A^*F_{M_1} = F_{M_2}$,
• $A^*w_2(M_1) = w_2(M_0)$$A^*w_2(M_1) = w_2(M_0)$,
• $A^*p_1(M_0) = p_1(M_1)$$A^*p_1(M_0) = p_1(M_1)$ and
• $A^*\KS(M_0) = A^*\KS(M_1)$$A^*\KS(M_0) = A^*\KS(M_1)$,

then there is a homeomorphism $f : M_0 \cong M_1$$f : M_0 \cong M_1$ inducing $A$$A$ on $H^2$$H^2$. If, in addition, $\KS(M_0) = 0 = \KS(M_1)$$\KS(M_0) = 0 = \KS(M_1)$, then $f$$f$ may be chosen to be a diffeomorphism and $M_0 \cong M_1$$M_0 \cong M_1$ admits a unique smooth structure.

5 Further discussion

...

6 References

• [Jupp1973] P. E. Jupp, Classification of certain $6$$6$-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
• [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
• [Okonek&Van de Ven1995] C. Okonek and A. Van de Ven, Cubic forms and complex $3$$3$-folds, Enseign. Math. (2) 41 (1995), no.3-4, 297–333. MR1365849 (97b:32035) Zbl 0869.14018
• [Smale1962] S. Smale, On the structure of $5$$5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain $6$$6$-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
• [Zhubr2000] A. V. Zhubr, Closed simply connected six-dimensional manifolds: proofs of classification theorems, Algebra i Analiz 12 (2000), no.4, 126–230. MR1793619 (2001j:57041)