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Equivariant Maps from Stiefel Bundles to Vector Bundles

Part of: Fiber spaces and bundles Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  01 June 2016

Mahender Singh
Mahender Singh*
Affiliation:
Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, Knowledge City, SAS Nagar (Mohali), Post Office Manauli, Punjab 140306, India ([email protected])
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Abstract

Let E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

Keywords

cohomological dimensioncontinuous cohomologyequivariant mapStiefel manifoldprojective Stiefel manifold

MSC classification

Secondary: 55M20: Fixed points and coincidences 55M35: Finite groups of transformations (including Smith theory) 55R91: Equivariant fiber spaces and bundles 55M10: Dimension theory 55R25: Sphere bundles and vector bundles
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 231 - 250
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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