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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?

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 $K$?

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 $K.$

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?

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?

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 $K.$

Can we replace the condition that $Y$ is simply connected by a weaker condition?

Precisely, is there still a canonical isomorphism $(**)$ if $Y$ is simple and $\pi_1(Y)$ is a $K$-vector space?

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 $K.$

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?