6-manifolds: 1-connected
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1 Introduction
Let be the set of diffeomorphism classes of closed oriented smooth simply-connected 6-manifolds .
Similarly, let be the set of homeomorphism classes of closed, oriented topological manifolds.
In this article we report on the calculation of and begun by [Smale1962], extended in [Wall1966] and [Jupp1973] and finally completed in [Zhubr2000]. We shall write for either or .
An excellent summary for the case where is torsion free may be found in [Okonek&Van de Ven1995, Section 1].
For the case where , see 6-manifolds: 2-connected.
2 Examples and constructions
We first present some familiar 6-manifolds.
-
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, the 6-sphere. -
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, the -fold connected sum of . - , the -fold connected sum of .
- , 3-dimensional complex projective space.
- , the non-trivial linear 4-sphere bundle over .
- For each we have , the corresponding 2-sphere bundle over . If we write 1 for a generator of then is diffeomorphic to .
- 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 be an -component framed link and let denote by the outcome of surgery on . Then is a simply connected spinable 6-manifold with and .
3 Invariants
The following gives a list of the key invariants needed to classify 1-connected 6-manifolds :
- The 3rd Betti-number, which is even since the intersection for of is skew-symmetric.
- The second Stiefel-Whitney class of is an element of which we regard as a homomorphism .
- The first Pontrjagin class .
- The Kirby-Siebenmann class .
- The cup product .
These invariants satisfy the following relation
for all which reduce to mod and for all which reduce to mod . As [Okonek&Van de Ven1995, p. 300] remark, in the smooth case this follows from the integrality of the -genus but in the topological case requires further arguments carried out in [Jupp1973].
Note that ifTex syntax errorthen the intersection form of is isomorphic to copies of , the skew-symmetric hyperbolic form on
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4 Classification
In this section we organise the classification results for simply-connected 6-manifolds.
4.1 Notation
Let be the set of isomorphism classes of pairs where is a finitely generated abelian group is a homomorphism and where an isomorphism is an isomorphism of groups commuting with the homomorphisms to . The second Stiefel-Whitney classes defines a surjection
and we let denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition
where ranges over all of .
4.2 Splitting Theorem
The following theorem of Wall reduces the classification of simply connected 6-manfiolds to the 2-connected case and the case where is torsion free.
Theorem 3 4.1 [Wall1966, Theorem 1].
Let be a closed, smooth, simply-connected 6-manifold withTex syntax error. Then up to diffeomorphism, there is a unique maniofld with such that is diffeomorphic to .
4.3 Smoothing theory
Let be a topological 6-manifold and recall the Kirby-Siebenmann invariant defined in [Kirby&Siebenmann1977]}: from the far reaching results of this book we have the following
Theorem 4.2. Let be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, is the sole obstruction to admitting a smooth structure.
H2 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 is torsion free.
Recall that the following system of invariants .
Theorem 4.3 [Jupp1973]. Let and be 1-connected 6-manifolds with torsion free. Suppose that is an isomorphism of abelian groups such that
- ,
- ,
- and
- ,
then there is a homeomorphism inducing on . If, in addition, , then may be chosen to be a diffeomorphism and admits a unique smooth structure.
5 Further discussion
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6 References
- [Jupp1973] P. E. Jupp, Classification of certain -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 -folds, Enseign. Math. (2) 41 (1995), no.3-4, 297–333. MR1365849 (97b:32035) Zbl 0869.14018
- [Smale1962] S. Smale, On the structure of -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 -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)