Prove for any homology theory ($H_*, \partial_*$) with values in R-mod that satisfies the dimension axiom there is an isomorphism $H_n(X,A)$ $\cong$ $\tilde H_n(X/A)$ where $A$ $\subset$ $X$ is a closed subspace which is a neighborhood deformation retract.

My approach was to use the Meyer-Vietoris- sequence for pushouts which gives us $H_n(X,A)$ $\cong$ $H_n(X/A, pt)$ for all n.

Furthermore I applied the long exact sequence for pairs to $(X/A, pt)$ and we get $H_n(X/A)$ $\cong$ $H_n(X/A, pt)$ for all n except n=1 and n=0. Also we have $\tilde H_n(X/A)$ $\cong$ $H_n(X/A)$ for all n except n = 0.

Now I am stuck for n=0 and n=1. Any hints?

  • $\begingroup$ Use the reduced LES for the pair $(X/A, pt)$. $\endgroup$ – iwriteonbananas Jun 20 '15 at 9:57
  • $\begingroup$ I don't get it, I don't know the definition of $\tilde H_n(X/A, pt)$. $\endgroup$ – Cosmare Jun 20 '15 at 12:29
  • $\begingroup$ $\tilde H_n(Y,B) = H_n(Y,B)$ by definition. $\endgroup$ – iwriteonbananas Jun 20 '15 at 14:32
  • $\begingroup$ Why? Even for $n = 0$? $\endgroup$ – Cosmare Jun 20 '15 at 14:39
  • $\begingroup$ Yes, for all $n$. It's the definition. I should've added $B\ \neq \emptyset$ above $\endgroup$ – iwriteonbananas Jun 20 '15 at 14:40

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