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I'm trying to write an expository manuscript on Bousfield-Kan's Fiber lemma and the relevant constructions. The proof of one needed theorem in the way seems to claim the following: if $F$ is a functor on $R$ modules such that $F(0)=0$ and suppose that for every free, connected simplicial $R$ module $M$ $F(M)$ is $k$ connected. Than the 2-fold cross effect functor $F_{2}(M,N)=Ker(F(M \oplus N) \rightarrow F(M) \oplus F(N))$ takes every connected simplicial free modules $M,N$ to a $k+1$ connected module. Is this true? The functors there are, of course, specific, but they are constructed inductively, each one built from cross effects on the one before him, so the "far" functors are less tangible. Regardless, if the above claim is true under certain assumptions, I'd be happy to know.

Another way to address the lemma there would be to ask: If $h:M \rightarrow RM$ is the Hurewicz homomorphism, and $F$ is a functor as above that increases connectivity by $k$, would $F(h)$ have the "Hurewicz property":it will induce an isomorphism $\pi_{i}(M) \cong \pi_{i}(RM)$ in the first dimension that the group on the left is not zero. Is this true under certain assumptions?

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  • $\begingroup$ You meant to write $\pi_iF(M)≅\pi_iF(RM)$, I assume. $\endgroup$ – Tom Goodwillie Jun 2 '11 at 19:55
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The first (seeming) claim is false for the functor $F(M)=M\otimes M$. If $M$ is connected then $F(M)$ is $1$-connected, but of course $F_2(M,M)=F(M)\oplus F(M)$ will never be any more highly connected than $F(M)$.

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