Mazur manifold

History

Barry Mazur and Valentin Poénaru discovered these manifolds simultaneously. Selman Akbulut and Robion Kirby showed that the Brieskorn homology spheres \Sigma(2,5,7), \Sigma(3,4,5), and \Sigma(2,3,13) are boundaries of Mazur manifolds, effectively coining the term `Mazur Manifold.' These results were later generalized to other contractible manifolds by Andrew Casson, John Harer, and Ronald Stern. One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds.

Mazur manifolds have been used by Ronald Fintushel and Stern to construct exotic actions of a group of order 2 on the 4-sphere.

Mazur's discovery was surprising for several reasons:

:* Every smooth homology sphere in dimension n \geq 5 is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Michel Kervaire and the h-cobordism theorem. Slightly more strongly, every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). But not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the Poincaré homology sphere does not bound such a 4-manifold because the Rokhlin invariant provides an obstruction.

:* The h-cobordism Theorem implies that, at least in dimensions n \geq 6 there is a unique contractible n-manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball D^n. It's an open problem as to whether or not D^5 admits an exotic smooth structure, but by the h-cobordism theorem, such an exotic smooth structure, if it exists, must restrict to an exotic smooth structure on S^4. Whether or not S^4 admits an exotic smooth structure is equivalent to another open problem, the smooth Poincaré conjecture in dimension four. Whether or not D^4 admits an exotic smooth structure is another open problem, closely linked to the Schoenflies problem in dimension four.

Mazur's observation

Let M be a Mazur manifold that is constructed as S^1 \times D^3 union a 2-handle. Here is a sketch of Mazur's argument that the double of such a Mazur manifold is S^4. M \times [0,1] is a contractible 5-manifold constructed as S^1 \times D^4 union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold S^1 \times S^3. So S^1 \times D^4 union the 2-handle is diffeomorphic to D^5. The boundary of D^5 is S^4. But the boundary of M \times [0,1] is the double of M.

References

References

1. Mazur, Barry. (1961). "A note on some contractible 4-manifolds". [[Annals of Mathematics]].
2. Poenaru, Valentin. (1960). "Les decompositions de l'hypercube en produit topologique". [[Bulletin de la Société Mathématique de France]].
3. (1979). "Mazur manifolds". [[Michigan Mathematical Journal]].
4. (1981). "Some homology lens spaces which bound rational homology balls". [[Pacific Journal of Mathematics]].
5. Fickle, Henry Clay. (1984). "Knots, $\backslash Z$-homology 3-spheres and contractible 4-manifolds". Houston Journal of Mathematics.
6. Stern, Ronald. (1978). "Some Brieskorn spheres which bound contractible manifolds". [[Notices of the American Mathematical Society]].
7. Akbulut, Selman. (1991). "A fake compact contractible 4-manifold". [[Journal of Differential Geometry]].
8. (1981). "An exotic free involution on $S^\{4\}$". [[Annals of Mathematics]].
9. Kervaire, Michel A.. (1969). "Smooth homology spheres and their fundamental groups". [[Transactions of the American Mathematical Society]].