Mazur manifold explained

In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold-with-boundary which is not diffeomorphic to the standard 4-ball. Usually these manifolds are further required to have a handle decomposition with a single

1 -handle, and a single 2 -handle; otherwise, they would simply be called contractible manifolds. The boundary of a Mazur manifold is necessarily a homology 3-sphere.

History

\Sigma( 2 , 5 , 7 ) , \Sigma( 3 , 4 , 5 ) and \Sigma( 2 , 3 , 13 ) are boundaries of Mazur manifolds, effectively coining the term `Mazur Manifold.'^[3] These results were later generalized to other contractible manifolds by Casson, Harer and Stern.^{[4] [5] [6]} One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds.^[7]

Mazur manifolds have been used by Fintushel and Stern^[8] 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 Kervaire^[9] 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 Rochlin 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} x 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 x [ 0 , 1 ] is a contractible 5-manifold constructed as S ^{1} x 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} x S ^{3} . So S ^{1} x D ^{4} union the 2-handle is diffeomorphic to D ^{5} . The boundary of D ^{5} is S ^{4} . But the boundary of M x [ 0 , 1 ] is the double of M .

References

1. Mazur . Barry . A note on some contractible 4-manifolds . . 73 . 1 . 1961 . 221–228 . 0125574 . 10.2307/1970288. 1970288 .
2. Valentin . Poenaru . Les decompositions de l'hypercube en produit topologique . Bull. Soc. Math. France . 88 . 1960 . 113–129 . 0125572 . 10.24033/bsmf.1546 . free .
3. Selman . Akbulut . Robion . Kirby . Mazur manifolds . Michigan Math. J. . 26 . 1979 . 259–284 . 3 . 0544597 . 10.1307/mmj/1029002261. free .
4. Andrew . Casson . John L. . Harer . Some homology lens spaces which bound rational homology balls . 0634760 . . 96 . 1 . 1981 . 23–36 . 10.2140/pjm.1981.96.23 . free .
5. Henry Clay . Fickle . Knots, Z-Homology 3-spheres and contractible 4-manifolds . 467–493 . Houston J. Math. . 10 . 4 . 1984 . 0774711.
6. R.Stern . Some Brieskorn spheres which bound contractible manifolds . Notices Amer. Math. Soc. . 25 . 1978.
7. Selman . Akbulut . A fake compact contractible 4-manifold . . 33 . 1991 . 335–356 . 2 . 1094459 . 10.4310/jdg/1214446320 . free .
8. Fintushel . Ronald . Stern . Ronald J. . An exotic free involution on S ^{4} . . 113 . 1981 . 2 . 357–365 . 0607896 . 10.2307/2006987. 2006987 .
9. Kervaire . Michel A. . Smooth homology spheres and their fundamental groups . . 144 . 1969 . 67–72 . 0253347 . 10.1090/S0002-9947-1969-0253347-3. free .