Mazur manifold

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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[edit]

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[edit]

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[edit]

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[edit]

  1. ^ Mazur, Barry A note on some contractible $4$-manifolds. Ann. of Math. (2) 73 1961 221--228.
  2. ^ Valentin Poenaru, Les decompositions de l'hypercube en produit topologique, Bull. Soc. Math. France 88 (1960), 113-129.
  3. ^ S.Akbulut, R.Kirby, "Mazur manifolds," Michigan Math. J. 26 (1979), 259--284.
  4. ^ A.Casson, J.Harer, "Some homology lens spaces which bound rational homology balls." Pacific. J. Math. Vol 96, No 1, (1981) 23–36.
  5. ^ H.Fickle, "Knots, Z-Homology 3-spheres and contractible 4-manifolds," pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).
  6. ^ R.Stern,"Some Brieskorn spheres which bound contractible manifolds," Notices Amer. Math. Soc 25 (1978), A448.
  7. ^ Akbulut cork
  8. ^ S.Akbulut, "A Fake compact contractible 4-manifold" , Journ. of Diff. Geom. 33, (1991), 335-356
  9. ^ Fintushel, Ronald; Stern, Ronald J. An exotic free involution on $S^{4}$. Ann. of Math. (2) 113 (1981), no. 2, 357--365.
  10. ^ Kervaire, Michel A. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 1969 67--72.