Schuette–Nesbitt formula: Difference between revisions

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In [[differential topology]], a branch of mathematics, a '''Mazur manifold''' is a contractible, [[compact space|compact]], smooth 4-dimensional [[manifold]] (with boundary) which is not [[diffeomorphism|diffeomorphic]] to the standard [[4-ball]]. The boundary of a Mazur manifold is necessarily a [[homology sphere|homology 3-sphere]]. 
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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 <math>S^1 \times D^3</math> union a 2-handle. An observation of Mazur's shows that the [[Double (manifold)|double]] of such manifolds is [[Diffeomorphism|diffeomorphic]] to <math>S^4</math> with the standard smooth structure.
 
==Some properties==
In general the [[double (manifold)|double]] of a Mazur manifold is a [[homotopy sphere|homotopy 4-sphere]], thus such manifolds are a source of possible counter-examples to the smooth [[Generalized Poincaré Conjecture|Poincaré conjecture in dimension 4]].
 
== History ==
 
[[Barry Mazur]] <ref>Mazur, Barry A note on some contractible $4$-manifolds. Ann. of Math. (2)  73  1961 221--228.</ref> and Valentin Poenaru<ref>Valentin Poenaru, Les decompositions de l'hypercube en produit topologique, Bull. Soc. Math. France 88 (1960), 113-129.</ref> discovered these manifolds simultaneously. Akbulut and Kirby showed that the [[homology sphere|Brieskorn homology spheres]] <math>\Sigma(2,5,7) </math>,   <math> \Sigma(3,4,5)</math> and <math>\Sigma(2,3,13)</math> are boundaries of Mazur manifolds.<ref>S.Akbulut, R.Kirby, "Mazur manifolds," Michigan Math. J. 26 (1979), 259--284.</ref> This results were later generalized to other contractible manifolds by Casson, Harer and Stern.<ref>A.Casson, J.Harer, "Some homology lens spaces which bound rational homology balls." Pacific. J. Math. Vol 96, No 1, (1981) 23&ndash;36.</ref><ref>H.Fickle, "Knots, Z-Homology 3-spheres and contractible 4-manifolds," pp. 467--493, Houston J. Math. Vol 10, No. 4 (1984).</ref><ref>R.Stern,"Some Brieskorn spheres which bound contractible manifolds," Notices Amer. Math. Soc 25 (1978), A448.</ref> One of the Mazur manifolds is also an example of an Akbulut cork <ref>http://en.wikipedia.org/wiki/Akbulut_cork</ref> which can be used to construct exotic 4-manifolds.<ref>S.Akbulut, "A Fake compact contractible 4-manifold" , Journ. of Diff. Geom. 33, (1991), 335-356</ref>
 
Mazur manifolds have been used by Fintushel and Stern <ref>Fintushel, Ronald; Stern, Ronald J. An exotic free involution on $S^{4}$. Ann. of Math. (2) 113 (1981), no. 2, 357--365.</ref> to construct exotic actions of a group of order 2 on the [[n-sphere|4-sphere]].
 
Mazur's discovery was surprising for several reasons:
 
:* Every smooth homology sphere in dimension <math>n \geq 5</math> is homeomorphic to the boundary of a compact contractible smooth manifold. This follows from the work of Kervaire <ref>Kervaire, Michel A. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144  1969 67--72.</ref> and the [[h-cobordism]] theorem.  Every smooth homology 4-sphere is diffeomorphic to the boundary of a compact contractible smooth 5-manifold (also by the work of Kervaire). Moreover, not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold. For example, the [[Homology sphere|Poincaré homology sphere]] does not bound such a 4-manifold because the [[Rokhlin invariant|Rochlin invariant]] provides an obstruction.
 
:* The [[h-cobordism|H-cobordism Theorem]] implies that, at least in dimensions <math>n \geq 6</math> there is a unique contractible <math>n</math>-manifold with simply-connected boundary, where uniqueness is up to diffeomorphism. This manifold is the unit ball <math>D^n</math>. It's an open problem as to whether or not <math>D^5</math> 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 <math>S^4</math>. Whether or not <math>S^4</math> admits an exotic smooth structure is equivalent to another open problem, the smooth [[Generalized Poincaré conjecture|Poincaré conjecture in dimension four]]. Whether or not <math>D^4</math> admits an exotic smooth structure is another open problem, closely linked to the [[Schoenflies problem]] in dimension four.
 
== Mazur's Observation ==
 
Let <math>M</math> be the Mazur manifold, constructed as <math>S^1 \times D^3</math> union a 2-handle. Here is a sketch of Mazur's argument that the [[Double (manifold)|double]] of such a Mazur manifold is <math>S^4</math>. <math>M \times [0,1]</math> is a contractible 5-manifold constructed as <math>S^1 \times D^4</math> union a 2-handle. The 2-handle can be unknotted since the attaching map is a framed knot in the 4-manifold <math>S^1 \times S^3</math>. So <math>S^1 \times D^4</math> union the 2-handle is diffeomorphic to <math>D^5</math>. The boundary of <math>D^5</math> is <math>S^4</math>. But the boundary of <math>M \times [0,1]</math> is the [[Double (manifold)|double]] of <math>M</math>.
 
== References ==
 
<references/>
 
[[Category:Differential topology]]
[[Category:Manifolds]]

Latest revision as of 01:00, 12 November 2014

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