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 ${\displaystyle S^{1}\times D^{3}}$ 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.

History

Barry Mazur^[1] and Valentin Poenaru^[2] discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres ${\displaystyle \Sigma (2,5,7)}$, ${\displaystyle \Sigma (3,4,5)}$ and ${\displaystyle \Sigma (2,3,13)}$ 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 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 ${\displaystyle D^{5}}$ 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 ${\displaystyle S^{1}\times D^{3}}$ 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 ${\displaystyle S^{1}\times D^{4}}$ union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold ${\displaystyle S^{1}\times S^{3}}$. So ${\displaystyle S^{1}\times D^{4}}$ union the 2-handle is diffeomorphic to ${\displaystyle D^{5}}$. The boundary of ${\displaystyle D^{5}}$ is $S^4$. But the boundary of $M \times [0,1]$ is the double of $M$.

References

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