Simplyconnected manifolds with a given boundary
Author:
Steven Boyer
Journal:
Trans. Amer. Math. Soc. 298 (1986), 331357
MSC:
Primary 57N13; Secondary 57N10
MathSciNet review:
857447
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Abstract: Let be a closed, oriented, connected manifold. For each bilinear, symmetric pairing , our goal is to calculate the set of all oriented homeomorphism types of compact, connected, oriented manifolds with boundary and intersection pairing isomorphic to . For each pair which presents , we construct a double coset space and a function . The set is the quotient of the group of all linkpairing preserving isomorphisms of the torsion subgroup of by two naturally occuring subgroups. When is an even pairing, we construct another double coset space , a function and a projection such that . Our main result states that when is even the function is injective, as is the function when is odd. Here is a KirbySiebenmann obstruction to smoothing. It follows that the sets are finite and of an order bounded above by a constant depending only on . We also show that when and is even, is injective. It seems likely that via the functions and , the sets and calculate when is respectively odd and even. We verify this in several cases, most notably when is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two connected manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an , depending only on , such that for any selfhomeomorphism of , extends to a selfhomeomorphism of any connected, compact manifold with boundary .
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 [G,L]
 C. Gordon and R. Litherland, On the signature of a link, Invent. Math. 47 (1978), 5369. MR 0500905 (58:18407)
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 S. Kaplan, Constructing framed manifolds with given almost framed boundaries, Trans. Amer. Math. Soc. 254 (1979), 237263. MR 539917 (82h:57015)
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 J. Milnor, Spin structures on manifolds, Enseign. Math. 9 (1963), 198203. MR 0157388 (28:622)
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 J. Morgan, Letter to R. Kirby, 1975.
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 F. Quinn, Ends of maps. III, Dimensions 4 and 5, J. Differential Geom. 17 (1982), 503521. MR 679069 (84j:57012)
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 D. Rolfsen, Knots and links, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
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 L. Taylor, Relative Rochlin invariants, Topology Appl. 18 (1984), 259280. MR 769295 (86g:57027)
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DOI:
http://dx.doi.org/10.1090/S00029947198608574476
PII:
S 00029947(1986)08574476
Article copyright:
© Copyright 1986
American Mathematical Society
