New York Journal of Mathematics
Volume 18 (2012) 59-74


Andrew Baker

On the cohomology of loop spaces for some Thom spaces

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Published: February 12, 2012
Keywords: Thom space, loop space, Eilenberg-Moore spectral sequence
Subject: primary 55P35; secondary 55R20, 55R25, 55T20

In this paper we identify conditions under which the cohomology H*(ΩMξ;k) for the loop space ΩMξ of the Thom space Mξ of a spherical fibration ξ\downarrow B can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class e(ξ)∈ Hn(B;k) vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum ΣΩMξ has a local splitting replacing the James splitting of ΣΩMξ when Mξ is a suspension.

Author information

School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QW, Scotland.