Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 752493
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57Q10, 18F25, 57R67

Retrieve articles in all journals with MSC: 57Q10, 18F25, 57R67


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0752493-3
PII: S 0002-9947(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