# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] DOI: 10.2307/1970382 · Zbl 0152.21902 [2] DOI: 10.1093/qmath/5.1.271 · Zbl 0057.39301 [3] DOI: 10.2307/1970107 · Zbl 0090.12905 [4] Mcller, Pacific J. Math. 116 pp 143– (1985) · Zbl 0569.55010 [5] Moller, On the cohomology of spaces of sections of real projective bundles (1983) [6] DOI: 10.2307/2045324 · Zbl 0514.55011 [7] James, General Topology and Homotopy Theory (1984) [8] DOI: 10.2307/1993204 · Zbl 0084.39002 [9] DOI: 10.1090/S0002-9904-1969-12266-4 · Zbl 0185.27202 [10] Hilton, Localization of nilpotent spaces (1975) · Zbl 0327.55012 [11] Hilton, Homotopy theory and duality (1967) [12] DOI: 10.2307/1998494 · Zbl 0479.55007 [13] Dold, Proc. Int. Conf. on Geometric Topology pp 81– (1980) [14] Crabb, Math. Scand. 55 pp 67– (1984) · Zbl 0581.55004 [15] Crabb, Z/-Homotopy Theory (1980) [16] DOI: 10.1093/qmath/34.2.223 · Zbl 0525.55006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.