Eilenberg-Moore spectral sequence calculation of function space cohomology
Computing the cohomology of a function space is difficult. The tool developed in this paper is the Eilenberg-Moore spectral sequence which arises from applying the function space functor \({\mathcal F}(-,Z)\) to a pushout diagram for \(\alpha: A\to Y\), and \(i: A\hookrightarrow X\), \[ \begin{matrix} Y\cup_\alpha X& \leftarrow & X&& {\mathcal F}(Y\cup_\alpha X,Z) & \rightarrow & {\mathcal F}(X,Z)\\ \uparrow && \uparrow&\Longrightarrow&\downarrow && \downarrow\\ Y &\leftarrow &A&&{\mathcal F}(Y,Z) &\rightarrow &{\mathcal F}(A,Z)\end{matrix} \] to obtain a pullback of function spaces. If you assume that the injection \(i: A\hookrightarrow X\) induces a surjection of function spaces \(i^{\#}: {\mathcal F}(X,Z) \to {\mathcal F}(A,Z)\), and that \({\mathcal F}(A,Z)\) is simply-connected and of finite type, then there is an Eilenberg-Moore spectral sequence converging to \(H^*({\mathcal F}(Y\cup_\alpha X); R)\) with \(E_2\)-term given as \[ E_2^{\ast,\ast} \cong \text{Tor}_{H^*({\mathcal F}(A,Z);R)}(H^*({\mathcal F}(X,Z);R), H^*({\mathcal F}(Y,Z);R)), \] and this is a spectral sequence of algebras over \(R\) a commutative Noetherian ring. When the space \(A\) is a sphere, \(i: S^{m-1}\hookrightarrow D^m\) and \(\alpha : S^{m-1} \to Y\), then the new space \(Y\cup_\alpha D^m\) is a cellular adjunction space, and the basis for induction arguments. In particular, surfaces can be constructed by adjoining disks to circles and so this method allows one to compute the cohomology of \({\mathcal F}(S, BG)\) where \(S\) is a surface and \(BG\) is the classifying space of a simply-connected Lie group that is \(p\)-torsion free. This space is relevant to the study of gauge theory. The input includes the cohomology of free loop spaces, well developed by the author. The use of this particular version of the Eilenberg-Moore spectral sequence is virtuosic and exploits particular resolutions, rational homotopy theory, and classical collapse results. Many of the results are technical, but one particularly interesting result is the fact that two surfaces are homeomorphic if and only if the function spaces \({\mathcal F}(S_i, BG)\) have the same homotopy after \(p\)-completion when \(p\) is an odd prime and \(G\) a non-trivial simply-connected Lie group whose homology is \(p\)-torsion free.
scientific article; zbMATH DE number 2105157