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

Barry Mazur[1] and Valentin Poénaru[2] 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.'[3] These results were later generalized to other contractible manifolds by Andrew Casson, John Harer, and Ronald 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 Ronald 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 Michel 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 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

  1. ^ Mazur, Barry (1961). "A note on some contractible 4-manifolds". Annals of Mathematics. 73 (1): 221–228. doi:10.2307/1970288. JSTOR 1970288. MR 0125574.
  2. ^ Poenaru, Valentin (1960). "Les decompositions de l'hypercube en produit topologique". Bulletin de la Société Mathématique de France. 88: 113–129. doi:10.24033/bsmf.1546. MR 0125572.
  3. ^ Akbulut, Selman & Kirby, Robion (1979). "Mazur manifolds". Michigan Mathematical Journal. 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 Journal of Mathematics. 96 (1): 23–36. doi:10.2140/pjm.1981.96.23. MR 0634760.
  5. ^ Fickle, Henry Clay (1984). "Knots, \Z-homology 3-spheres and contractible 4-manifolds". Houston Journal of Mathematics. 10 (4): 467–493. MR 0774711.
  6. ^ Stern, Ronald (1978). "Some Brieskorn spheres which bound contractible manifolds". Notices of the American Mathematical Society. 25
  7. ^ Akbulut, Selman (1991). "A fake compact contractible 4-manifold". Journal of Differential Geometry. 33 (2): 335–356. doi:10.4310/jdg/1214446320. MR 1094459.
  8. ^ Fintushel, Ronald & Stern, Ronald J. (1981). "An exotic free involution on S^{4}". Annals of Mathematics. 113 (2): 357–365. doi:10.2307/2006987. JSTOR 2006987. MR 0607896.
  9. ^ Kervaire, Michel A. (1969). "Smooth homology spheres and their fundamental groups". Transactions of the American Mathematical Society. 144: 67–72. doi:10.1090/S0002-9947-1969-0253347-3. MR 0253347.
  • Rolfsen, Dale (1990), Knots and links. Corrected reprint of the 1976 original., Vol. 7, Mathematics Lecture Series, Houston, TX: Publish or Perish, Inc., pp. 355–357, Chapter 11E, ISBN 0-914098-16-0. MR 1277811