A decomposition theorem for h -cobordant smooth simply-connected compact 4-manifolds

Curtis, Cynthia L.; Freedman, Michael; Hsiang, W. C.; Stong, Richard

doi:10.1007/s002220050031

Cited by 46 publications

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“…Weaker (and more technical) forms of theorems 1 and 2 can be proven using the same techniques for 1-connected, smooth 4-manifolds with boundary. A result similar to theorem 2 appears as part of the author's contribution to [2].…”

Section: Theorem 1 Suppose M Is a Closed Simply-connected Smooth 4mentioning

confidence: 58%

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

Stong

1995

Mathematical Research Letters

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A b s t r a c t . We will prove that any closed, simply-connected smooth 4-manifold admits a handlebody structure where the 2-handles homotopically cancel the 1-handles and the dual 1-handles in the nicest possible way. As a consequence we will derive the following improved splitting theorem for closed, simply-connected smooth 4-manifolds. Suppose M is a closed, simply-connected, smooth 4-manifold and the intersection form of M splits as (where M i is a compact, simply-connected, smooth 4-manifold with boundary the homology 3-sphere Σ and intersection form isomorphic to (Z n i , λ i ). This result is also extended to 4-manifolds with free fundamental groups.

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Section: Theorem 1 Suppose M Is a Closed Simply-connected Smooth 4mentioning

confidence: 58%

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

Stong

1995

Mathematical Research Letters

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“…If X o is homeomorphic to Xl then there is a compact contractible 4-manifold W C Xo such that by cutting W out of X o and regluing it by an involution on its boundary, we get a smooth 4-manifold diffeomorphic to Xl, see [62] and [211]. If X o is homeomorphic to Xl then there is a compact contractible 4-manifold W C Xo such that by cutting W out of X o and regluing it by an involution on its boundary, we get a smooth 4-manifold diffeomorphic to Xl, see [62] and [211].…”

Section: Constructing Smooth Manifoldsmentioning

confidence: 99%

Invariants for Homology 3-Spheres

Saveliev

2002

Encyclopaedia of Mathematical Sciences

122

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“…Later, a comprehensive analysis of 1-connected h-cobordisms extended Akbulut's result (see [8,13,12]). The following picture of the general 1-connected 5-dimensional h-cobordism (W ; P, Q) emerges.…”

Section: -Manifold Pairingsmentioning

confidence: 84%

Universal manifold pairings and positivity

et al. 2005

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Gluing two manifolds M 1 and M 2 with a common boundary S yields a closed manifold M . Extending to formal linear combinations x = Σa i M i yields a sesquilinear pairing p = , with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a "unitary" TQFT can potentially detect any nontrivial x, we ask if x, x = 0 whenever x = 0. If this is the case, we call the pairing p positive. The question arises for each dimension d = 0, 1, 2, . . .. We find p(d) positive for d = 0, 1, and 2 and not positive for d = 4. We conjecture that p(3) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly s-cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for d = 3 + 1. There is a further physical implication of this paper. Whereas 3-dimensional ChernSimons theory appears to be well-encoded within 2-dimensional quantum physics, e.g. in the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero. AMS Classification numbers Primary: 57R56, 53D45Secondary: 57R80, 57N05, 57N10, 57N12, 57N13

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A decomposition theorem for h -cobordant smooth simply-connected compact 4-manifolds

Cited by 46 publications

References 0 publications

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds

Invariants for Homology 3-Spheres

Universal manifold pairings and positivity

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