Mazur manifold
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In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold (which is not diffeomorphic to the standard 4-ball) whose boundary is a homology 3-sphere. The double of such a manifold is a homotopy 4-sphere, thus such manifolds are a source of possible counter-examples to the smooth Poincare conjecture in dimension 4. Mazur manifolds have also been used by Fintushel and Stern to construct exotic actions of a group of order 2 on the 4-sphere.
Barry Mazur gave the first example of such manifolds. He showed that the Brieskorn homology sphere Σ(2,5,7) is the boundary of a contractible 4-manifold. His results were later generalized by Kirby, Akbulut, Casson, Harer and Stern.
Frequently the term Mazur manifold is restricted to a special class of the above -- 4-manifolds that have a handle decomposition containing 3 handles: a single 0-handle, 1-handle and 2-handle. The reason this restriction is interesting is that the double of such a 4-manifold is known to be diffeomorphic to the 4-sphere.
[edit] References
- S.Akbulut, R.Kirby, "Mazur manifolds," Michigan Math. J. 26 (1979), 259--284.
- A.Casson, J.Harer, "Some homology lens spaces which bound rational homology balls." Pacific. J. Math. Vol 96, No 1, (1981) 23–36.
- H.Fickle, "Knots, Z-Homology 3-spheres and contractible 4-manifolds," pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).
- R.Fintushel, R.Stern, "An exotic free involution on S^4," Ann. Math. (2) 113 (1981) no2, 357--365.
- B.Mazur, "A note on some contractible 4-manifolds", Annals of Mathematics, (2) 73 (1961). 221–228.
- R.Stern,"Some Brieskorn spheres which bound contractible manifolds," Notices Amer. Math. Soc 25 (1978), A448.
