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 Smale, extended by Wall and Jupp and completed by Zhubr. We shall write for either or .
2 Examples and constructions
We first present some familiar 6-manifolds.
- , the standard 6-sphere.
- , 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 .
Surgery on framed links. Let be a framed link. Then , the outcome of surgery on , is a simply connected Spinable 6-manifold with and .
- ??? Complete intersections of some form.
1 Invariants
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 .
2 Classification
2.1 Preliminaries
Tex syntax error. 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 .
2.1 The splitting Theorem
Theorem 3 8.1 (Wall). Let be a closed, smooth, simply-connected 6-manifold with . Then up to diffeomorphism, there is a unique maniofld with such that is diffeomorphic to .
2.2 Smoothing theory
Theorem 1 8.2. Let be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class, is the sole obstruction to admitting a smooth structure.
Theorem 2 8.3. Every homeomorphism of simply-connected, smooth -manifolds is topologically isotopic to a diffeomorphism. Hence we have an injection
3 6-manifolds with torsion free second homology
4 2-connected 6-manifolds
Smale showed that every smooth, 2-connected 6-manifold is diffeomorphic to or a connected sum . Hence if denotes the third Betti-number of and denotes the natural numbers we obtain a bijection
Applying Theorems 1 and 2 we see that the same statement holds for .
5 Further discussion
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References
- [Jupp1973] P. E. Jupp, Classification of certain -manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [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