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- For the more general case where , see 6-manifolds: 1-connected.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
- For the more general case where , see 6-manifolds: 1-connected.
1 Introduction
Tex syntax errorbe the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
Tex syntax error. The classification
Tex syntax errorwas one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The classification, as for oriented surfaces is strikingly simple: every 2-connected 6-manifold
Tex syntax erroris diffeomorphic to a connected-sum
Tex syntax error
Tex syntax errorand in general
Tex syntax erroris determined by the formula for the Euler characteristic of
Tex syntax error
Tex syntax error
- For the more general case where
Tex syntax error
, see 6-manifolds: 1-connected.
1 Construction and examples
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
-
Tex syntax error
, the standard 6-sphere. -
Tex syntax error
, theTex syntax error
-fold connected sum ofTex syntax error
.
2 Invariants
Suppose thatTex syntax erroris diffeomorphic to
Tex syntax errorthen:
-
Tex syntax error
, - the third Betti-number of
Tex syntax error
is given byTex syntax error
, - the Euler characteristic of
Tex syntax error
is given byTex syntax error
, - the intersection form of
Tex syntax error
is isomorphic to the sum of b-copies ofTex syntax error
, the standard skew-symmetric hyperbolic form onTex syntax error
.
3 Classification
Recall that the following theorem was stated in other words in the introduction:
Theorem 7.1 [Smale1962a, Corolary 1.3].
The semi-group of 2-connected 6-manifolds is generated byTex syntax error.
Tex syntax errordenotes the natural numbers we obtain a bijection
Tex syntax error
4 Further discussion
4.1 Topological 2-connected 6-manifolds
LetTex syntax errorbe the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 9.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection
Tex syntax error
Proof.
For any such manifoldTex syntax errorwe have
Tex syntax errorand so
Tex syntax erroris smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 7.1 are diffeomorphic.
4.2 Mapping class groups
...
2 References
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103