Mazur manifold

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. The boundary of a Mazur manifold is necessarily a homology 3-sphere.

Frequently the term Mazur manifold is restricted to a special class of the above definition: 4-manifolds that have a handle decomposition containing exactly three handles: a single 0-handle, a single 1-handle and single 2-handle. This is equivalent to saying the manifold must be of the form $S^1 \times D^3$ union a 2-handle. An observation of Mazur's shows that the double of such manifolds is diffeomorphic to S⁴ with the standard smooth structure.

Some properties

In general the double of a Mazur manifold is a homotopy 4-sphere, thus such manifolds are a source of possible counter-examples to the smooth Poincare conjecture in dimension 4.

History

Barry Mazur ^[1] 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. ^{[2] [3] [4] [5]}

Mazur manifolds have been used by Fintushel and Stern ^[6] to construct exotic actions of a group of order 2 on the 4-sphere.

Mazur's discovery was something of a surprise for several reasons:

• Every homology sphere in dimension $n \geq 5$ bounds a contractible manifold. This follows from the work of Kervaire ^[7] and the h-cobordism theorem. Every homology 4-sphere bounds a contractible 5-manifold (also by Kervaire). Moreover, not every homology 3-sphere bounds a contractible 4-manifold. For example, the Poincare homology sphere does not bound a contractible 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 Dⁿ. It's an open problem as to whether or not D⁵ 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 S⁴. Whether or not S⁴ admits an exotic smooth structure is equivalent to another open problem, the smooth Poincare conjecture in dimension four. Whether or not D⁴ admits an exotic smooth structure is another open problem, closely linked to the Schoenflies problem in dimension four.

Mazur's Observation

Let M be the Mazur manifold, 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 S⁴. $M \times [0,1]$ 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 D⁵. The boundary of D⁵ is S⁴. But the boundary of $M \times [0,1]$ is the double of M.

References

1. ^ Mazur, Barry A note on some contractible $4$-manifolds. Ann. of Math. (2) 73 1961 221--228.
2. ^ S.Akbulut, R.Kirby, "Mazur manifolds," Michigan Math. J. 26 (1979), 259--284.
3. ^ A.Casson, J.Harer, "Some homology lens spaces which bound rational homology balls." Pacific. J. Math. Vol 96, No 1, (1981) 23–36.
4. ^ H.Fickle, "Knots, Z-Homology 3-spheres and contractible 4-manifolds," pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).
5. ^ R.Stern,"Some Brieskorn spheres which bound contractible manifolds," Notices Amer. Math. Soc 25 (1978), A448.
6. ^ Fintushel, Ronald; Stern, Ronald J. An exotic free involution on $S^{4}$. Ann. of Math. (2) 113 (1981), no. 2, 357--365.
7. ^ Kervaire, Michel A. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 1969 67--72.