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Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let $\mathrm{Cotor}_A(M,N)$ denote the cotorsion product of $M$ and $N$ relative to $A$.

The graded $K$-vector space $\mathrm{Cotor}_A(M, \, N)$ is by definition the homology of the totalization of the cosimplicial cochain complex over $K$ with $n$-th term $M \otimes A^{\otimes n} \otimes N$, where the tensor product is in cochain complexes over $K.$

Let $X \to Y$ be a Serre fibration between connected spaces and $F$ its fiber over a given point $y$ of $Y.$ If $Y$ is simply connected, by a theorem of Eilenberg and Moore there is a canonical isomorphism \begin{equation} H_*(F;K)\cong \mathrm{Cotor}_{C_*(Y; \, K)}(C_*(X; \, K), \, C_*(*; \, K)), \ \ \ \ (**) \end{equation} where $C_*(-;\, K)$ are singular chains with coefficients in the field $K.$

Question. Can we replace the condition that $Y$ is simply connected by a weaker condition? For example, is there still a canonical isomorphism $(**)$ if $$Y = BG = K(G,\, 1)$$ for $G$ a derived $p$-complete abelian group, where $p$ is the characteristic of $K$?

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  • $\begingroup$ In McCleary's book (p. 250, Theorem 7.14) states the result for $\operatorname{Tor}$ and assumes only that the system of local coefficients induced by the fibre on $B$ is simple. I couldn't find a dual statement in the book concerning Cotor, but even if there was one I guess that when $Y=BG$ the system may very well not be simple? $\endgroup$
    – Pedro
    Commented May 11, 2021 at 22:02
  • $\begingroup$ Thanks for your comment! Which book of McCleary do you mean? $\endgroup$
    – Hadrian Heine
    Commented May 12, 2021 at 20:36
  • $\begingroup$ Oh, I mean "User's Guide to Spectral Sequences." $\endgroup$
    – Pedro
    Commented May 12, 2021 at 20:46
  • $\begingroup$ In Theorem 7.14 of McCleary's book it is also assumed that the fiber $F$ of the fibration $X \to Y$ is connected to obtain the dual statement. I am especially interested in the case that $X$ is contractible, in which case $F$ is the loop space of $Y$. In this case $F$ is connected if and only if $Y$ is simply connected, which is what I like to drop. $\endgroup$
    – Hadrian Heine
    Commented May 12, 2021 at 22:32

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