 

Andrew Baker
On the cohomology of loop spaces for some Thom spaces view print


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


Abstract
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 EilenbergMoore spectral
sequence which has a particularly simple form when the Euler
class e(ξ)∈ H^{n}(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 Σ^{∞}ΩMξ
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.
a.baker@maths.gla.ac.uk

