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{{Stub}} == Introduction == <wikitex>; Let $\mathcal{M}_{6}$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 6-manifolds $M$. Similarly, let $\mathcal{M}^{\Top}_{6}$ be the set of [[wikipedia:Homeomorphism|homeomorphism]] classes of closed, oriented [[wikipedia:Topological_manifold|topological manifolds]]. In this article we report on the calculation of $\mathcal{M}^{}_{6}$ 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 $\mathcal{M}^{}_{6}$ or $\mathcal{M}^{\Top}_{6}$. 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. * $S^4 \times_\gamma S^2$, 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 $S^2 \times_1 S^4$ 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 $\phi \co \sqcup_r S^3 \times D^3 \to S^6$ be a framed link and let denote $M^6_\phi$ 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 $2$ and for all $K \in H^4(M)$ which reduce to $\KS(M)$ mod $2$. 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 forms|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 $$ w_2\co\mathcal{M}_{6}^{\Top} \longrightarrow \Hom({\mathcal Ab}, \Zz_2), \quad \quad [M] \longmapsto w_2 : H_2(M) \to \Zz/2$$ 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 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}} === H₂ torsion free === 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 $(b_3(M)/2, H^2(M), F_M, w_2(M), p_1(M), KS(M))$. <wikitex>; {{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]]

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