Mazur manifold

In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold (with boundary) 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 $S^4$ 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 Poincaré conjecture in dimension 4.

History

Barry Mazur^[1] and Valentin Poenaru^[2] discovered these manifolds simultaneously. Akbulut and 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.^[3] This 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^[7] which can be used to construct exotic 4-manifolds.^[8]

Mazur manifolds have been used by Fintushel and Stern^[9] 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^[10] 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 $S^1 \times D^3$ 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

↑ Mazur, Barry A note on some contractible $4$-manifolds. Ann. of Math. (2) 73 1961 221--228.
↑ Valentin Poenaru, Les decompositions de l'hypercube en produit topologique, Bull. Soc. Math. France 88 (1960), 113-129.
↑ 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.Stern,"Some Brieskorn spheres which bound contractible manifolds," Notices Amer. Math. Soc 25 (1978), A448.
↑ Akbulut cork
↑ S.Akbulut, "A Fake compact contractible 4-manifold" , Journ. of Diff. Geom. 33, (1991), 335-356
↑ Fintushel, Ronald; Stern, Ronald J. An exotic free involution on $S^{4}$. Ann. of Math. (2) 113 (1981), no. 2, 357--365.
↑ Kervaire, Michel A. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 1969 67--72.