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 [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.
-
, the 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
.
- 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 a framed link and let denote
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
![\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
. 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 if
then the intersection form of
is isomorphic to
copies of
, the skew-symmetric hyperbolic form on
.
4 Classification
4.1 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
.
4.2 The splitting Theorem
Theorem 3 4.1 [Wall1966, Theorem 1].
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
.
4.3 Smoothing theory
Theorem 4.2.
Let be a simply-connected, topological 6-manifold. The Kirby-Siebenmann class,
is the sole obstruction to
admitting a smooth structure.
6-manifolds with torsion free second homology
The pape [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 6.1 [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
...
6 References
- [Jupp1973] P. E. Jupp, Classification of certain
-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [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)