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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Wall invariant of certain $S^{1}$ bundles
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by Douglas R. Anderson PDF
Proc. Amer. Math. Soc. 31 (1972), 529-535 Request permission

Abstract:

Let $p:E \to B$ be a principal ${S^1}$ bundle with $B$ dominated by a finite complex. Then it is easy to show that $E$ is also dominated by a finite complex. In this paper we show, under suitable additional hypotheses, that in fact $E$ has the homotopy type of a finite complex. The proof is carried out by computing Wall’s finiteness obstruction for $E$.
References
    D. R. Anderson, The obstruction to the finiteness of the total space of a flat bundle (submitted). —, Whitehead torsions vanish in many ${S^1}$ bundles, Invent. Math. (to appear).
  • Stephen M. Gersten, A product formula for Wall’s obstruction, Amer. J. Math. 88 (1966), 337–346. MR 198465, DOI 10.2307/2373197
  • V. J. Lal, Wall obstruction of a fibration, Invent. Math. 6 (1968), 67–77. MR 231380, DOI 10.1007/BF01389834
    • W. S. Massey, On the fundamental group of certain fiber spaces, Ann. of Math. Studies, no. 53, Princeton Univ. Press, Princeton, N.J., 37-41.
  • J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
    • —, Uses of the fundamental group, Seventy-Third Summer Meeting of the Amer. Math. Soc., Colloquium Lectures, University of Wisconsin, Madison, Wis., 1968. L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. Thesis, Princeton University, Princeton, N.J., 1965.
  • C. T. C. Wall, Finiteness conditions for $\textrm {CW}$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR 171284, DOI 10.2307/1970382
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 31 (1972), 529-535
  • MSC: Primary 55.50
  • DOI: https://doi.org/10.1090/S0002-9939-1972-0287545-5
  • MathSciNet review: 0287545