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• W1996041982 abstract "Let p: E-B be a principal S' 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. Let X be a (possibly infinite) CW complex which is dominated by a finite complex and let A=Z(vr(X)). In [8] Wall defines an obstruction a(X)cKO(A) to finding a finite complex with the same homotopy type as X. If F i E B is a fiber space with total space, base, and fiber dominated by finite CW complexes, it is natural to ask whether a(E) can be computed in terms of a(F), a(B), and invariants of the fiber space. Since c(F), a(E), and o(B) are elements of KoZ(71(F)), kOZ(v1(E)), and koZ(v1(B)) respectively, and --17r(F)-v1(E)-v1(B)-... is exact, the solution of the general problem involves an extension problem. In an effort to obtain a better understanding of the relationship between these problems, we will prove the MAIN THEOREM. Let p: E-?B be a principal S1 bundle such that (i) B is a CW complex dominated by a finite complex; (ii) j#:1r(S1)-?--,1(E) is a monomorphism; and (iii) r1(E) is abelian. Then E is dominated by afinite complex and a(E)=O. The universal covering space EL of E and the action of r1(E) on P are determined in ?1. This information is then used in ?2 to prove the theorem. The problem considered here was first raised by Lal in [3]. Unfortunately, the main theorems of that paper (Theorems 2 and 3) do not hold in the generality Lal claims, but appear to require additional hypotheses. A forthcoming paper by the author [0] contains a counterexample to Lal's Theorems 2 and 3 as well as a modified version of Lal's Theorem 3. Received by the editors June 9, 1970 and, in revised form, April 23, 1971. AMS 1970 subject classifications. Primary 57C05; Secondary 57C10, 57C50. 'Partially supported by the NSF under grant number GP12837. ? American Mathematical Society 1972" @default.
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• W1996041982 date "1972-02-01" @default.
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• W1996041982 title "The Wall invariant of certain $S^{1}$ bundles" @default.
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• W1996041982 doi "https://doi.org/10.1090/s0002-9939-1972-0287545-5" @default.
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