On the cohomology of loop spaces for some Thom spaces (Q416776)

Let \(\xi\) be a spherical fibration with Thom complex \(M\xi\). The main result of the paper provides some conditions under which the \(\mathbb{F}_p\)-cohomology of the loop space \(\Omega M\xi\) is a polynomial algebra (Theorem 4.1 and Theorem 5.1) where \(p\) is a prime number. The author also identifies some conditions under which the cohomology ring is an exterior (\(p=2\)) or a truncated polynomial algebra (\(p>2\)). The proofs are based on Eilenberg-Moore spectral sequence arguments, and working with bar (cobar) resolutions. Moreover, this is a description of the rational cohomology of \(\Omega M\xi\).NEWLINENEWLINENEWLINE NEWLINEFinally, working in the category of \(S\)-modules by \textit{A. D. Elmendorf, I. Kříž, M. A. Mandell} and \textit{J. P. May} [Rings, modules, and algebras in stable homotopy theory. Mathematical Surveys and Monographs. 47. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0894.55001)], the author provides a local stable splitting for \(\Omega M\xi\) (Proposition 9.1) provided by a map NEWLINE\[NEWLINE\mathbb{T}(\Sigma^{-1}\Sigma^\infty M\xi)\longrightarrow \Sigma^\infty\Omega M\xiNEWLINE\]NEWLINE with \(\mathbb{T}\) being the free \(S\)-algebra functor. If \(M=\Sigma X\) then by definition of \(\mathbb T\) the above splitting looks like the stable form of James' splitting for \(\Omega\Sigma X\).NEWLINENEWLINENEWLINE NEWLINEThe paper includes background material on Thom complexes, Eilenberg-Moore spectral sequences, and the cohomology of sphere bundles.