3
$\begingroup$

For a reduced (co-)homology theory defined for the CW-complexes, is the following formulation equivalent to the axioms of excision and long exact sequence? It states that for a CW-pair $(X,A)$ there are boundary homomorphisms $\partial : h_n(X/A) \rightarrow h_{n-1}(A)$ that fit into an exact sequence $$\begin{equation} \cdots \rightarrow^{\partial} h_n(A) \rightarrow^{i_*} h_n(X) \rightarrow^{q_*} h_n(X/A) \rightarrow^{\partial} h_{n-1}(A) \rightarrow \cdots\end{equation}$$ where $i$ is the inclusion and $q$ the quotient map. It states also that the boundary map is natural, which means that given a map $f:(X, A) \rightarrow (Y, B)$ inducing a quotient map $\overline{f}:X/A \rightarrow Y/B$ the equality $$ \partial_X f_* = \overline{f}_*\partial_Y $$ holds. I know that the excision axiom and the long exact sequence of the pair imply this formulation, but I can't easily see why the converse is true. In particular, the excision is the one that I can't see how to prove. The long exact sequence I think follows easily from the isomorphism $$h_n(X,A) \simeq h_n(X/A, A/A)$$ and then we can put $h_n(X, A)$ in the long exact sequence.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.