Simply-connected -manifolds with a given boundary

Author:
Steven Boyer

Journal:
Trans. Amer. Math. Soc. **298** (1986), 331-357

MSC:
Primary 57N13; Secondary 57N10

DOI:
https://doi.org/10.1090/S0002-9947-1986-0857447-6

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 link-pairing 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 Kirby-Siebenmann 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 self-homeomorphism of , extends to a self-homeomorphism of any -connected, compact -manifold with boundary .

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0857447-6

Article copyright:
© Copyright 1986
American Mathematical Society