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The space of sections of a sphere-bundle. I. (English) Zbl 0614.55012
Let \(\xi\) be a real \((n+1)\)-plane bundle \((n>0)\) equipped with a Riemannian metric over a finite connected CW-complex X of dimension m. The authors study the topology of the space \(\Gamma\) \(S\xi\) of sections of the sphere bundle \(S\xi\) of \(\xi\). Among other results they prove that for \(n>2m+1\) the homotopy type of \(\Gamma\) \(S\xi\) determines the stable homotopy type of the (stable) Thom space \(X^{-\xi}.\)
Other results deal with the homology of \(\Gamma\) \(S\xi\). Let \({\mathcal H}\) be the group of orientation-preserving isometric isomorphisms of \(\xi\) over X and \(P\to B\) a principal \({\mathcal H}\)-bundle over a finite CW- complex B. The main result is a computation of the homology \(H_ j(P \times_{{\mathcal H}} \Gamma S\xi)\), up to group extension, for \(j<2n-2m- 1\). The proof uses a sort of ”Gysin sequence” involving B and \(X^{- \xi}\). As an application the authors prove among others two results of J. M. Møller on the homology of spaces of sections of projective bundles.
Reviewer: V.L.Hansen

MSC:
55R99 Fiber spaces and bundles in algebraic topology
55S40 Sectioning fiber spaces and bundles in algebraic topology
58D15 Manifolds of mappings
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