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Structure sets vanish for certain bundles over Seifert manifolds


Author: Christopher W. Stark
Journal: Trans. Amer. Math. Soc. 285 (1984), 603-615
MSC: Primary 57Q10; Secondary 18F25, 57R67
DOI: https://doi.org/10.1090/S0002-9947-1984-0752493-3
MathSciNet review: 752493
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Abstract: Let $ {M^{n + 3}}$ be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold $ {K^3}$ with an effective circle action of hyperbolic type. Assume that the fiber $ {N^n}$ in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set $ {S_{\text{TOP}}}\,({N^n} \times {D^k},\partial )$ of topological surgery vanishes for sufficiently large $ k$. Then the projective class and Whitehead groups of $ M$ vanish and $ {S_{\text{TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0$ if $ n + k \geqslant 3$ or if $ {K^3}$ is closed and $ n = 2$. The UNil groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that $ {\text{UNil}}_j$ vanishes in spite of the failure of Cappell's sufficient condition.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0752493-3
Keywords: Seifert manifold, fiber bundle, algebraic $ K$-theory, projective class group, Whitehead group, surgery, structure set, splitting theorems, UNil
Article copyright: © Copyright 1984 American Mathematical Society

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