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] in [Jupp1973] and completed in [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
![\displaystyle W^3 = (p_1(M) + 24K) \cup W](/images/math/8/f/a/8fa8a63919d453971d1cf6844e2ef4e8.png)
for all which reduce to
mod
and for all
which reduce to
mod
.
2 Classification
2.1 Preliminaries
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
![\displaystyle w\co\mathcal{M}_{6}^{\Top}(e) \rightarrow \Hom({\mathcal Ab}, \Zz_2)](/images/math/9/a/6/9a69c15041f0e38b7feb705f72c45cbc.png)
and we let denote the set of isomorphism classes of 6-manifolds with prescribed second Stiefel-Whitney class. We obtain the decomposition
![\displaystyle \mathcal{M}^{\Cat}_{6}(e) = \cup_{[G, w]} \mathcal{M}^{\Cat}_{6}(G, w)](/images/math/b/c/4/bc42f0b862f212b051befba26a41415e.png)
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
![\displaystyle \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).](/images/math/5/4/7/54760c11b8f0267445e9ec57732484ef.png)
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
![\displaystyle \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).](/images/math/4/f/8/4f87caab28b3b9b7fd16a5f88c9bb677.png)
Applying Theorems 1 and 2 we see that the same statement holds for .
5 Further discussion
6 References
- [Jupp1973] P. E. Jupp, Classification of certain
-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [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
- [Zhubr] Template:Zhubr