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If $X$ and $Y$ aren't 1-connected, then $f$ lifts to a map of universal covers $\tilde f: \tilde X \to \tilde Y$ and your assumption about local coefficients implies that $\tilde f$ is a homology isomorphism …

My Questions: (1) For what class of spaces $X$ does $H_*(X^+,+)$ coincide with definition of Borel-Moore homology given by locally finite chains? … (I suspect not, since ordinary singular homology in degree zero is always free abelian.) …

Then $F_\ast(X,Y)$ is a cosimplicial space and we can consider its homology spectral sequence. … Bousfield gave conditions for when this will converge to the homology of $F_\ast(X,Y)$ with field coefficients. See here: On the Homology Spectral Sequence of a Cosimplicial Space A. K. …

It's known that the inclusion of a meridian $S^1$ (a small circle which links the knot exactly once) into the knot complement $S^3 \setminus S^1$ is a homology isomorphism. … The inclusion of the meridian $S^1 \subset X$ is a homology isomorphism, a $\pi_1$ isomorphism but is never a weak equivalence since by a result of Levine if it were then the knot would be trivial (we …

The following question is related to my previous post on co-$A_\infty$ spaces (co-$A_\infty$ spaces), but goes in a somewhat different direction. Some Background: In trying to classify $A_\infty$ …

This gives a diagram $$ B \quad \overset{a}\leftarrow \quad E/F \quad \overset{b} \to \quad \Sigma F $$ and when the transgression is defined it is given by the homomorphism these maps induce on homology … Thus the lift is unique in this case (no indeterminacy) and the transgression is the map $\Sigma \Omega X \to X$ up to sign on homology. …