# Two questions on axiomatic homology

1) Given the Eilenberg-Steenrod axioms, there are several Mayer-Vietoris type sequences that can be deduced. The most general form seems to be

$$\rightarrow H_n(X \cap Y, A \cap B) \rightarrow H_n(X,A) \oplus H_n(Y,B) \rightarrow H_n(X \cup Y, A \cup B) \rightarrow H_{n-1}(X \cap Y, A \cap B) \rightarrow$$

where $A,B,X,Y$ are (say) open with $A \subseteq X, B \subseteq Y$.

Question: How can one define the connecting map for this sequence?

After playing around for a while I (!) could find neither a braid nor ladder sequence to produce it. The only general proof I know goes via mapping cylinder constructions and thus obscures this point somewhat.

2) Given an additive, multiplicative homology theory and a space $X$ with flat homology $H_\ast(X)$ the mulitplication gives an isomorphism $$H_\ast(X \times Y) \cong H_\ast(X) \otimes_{H_\ast(pt)} H_\ast(Y)$$ for $Y$ a cell complex.

Question: Is this also true for general $Y$?

(Of course it is if $H$ sends weak equivalences to isomorphisms)

• I'm a little confused. You seem to say that you know the proof of exactness. I don't see how this is possible if you don't also know the definition of the connecting map. – Steven Landsburg Oct 15 '12 at 19:07
• If $H$ sends weak equivalences to isomorphisms, any space has the homology of a CW-complex, so 2) holds for any space. – Fernando Muro Oct 15 '12 at 19:33
• @Steven Landsburg: Well, I know one definition via an iterated mapping cylinder construction, but it is a bit convoluted and not along the same lines as the usual construction for say the absolute version of the MVS. @Fernando Muro: That's what the comment in parentheses says, right?! – FJH Oct 16 '12 at 9:01
• Sorry, I must have misunderstood something. – Fernando Muro Oct 18 '12 at 11:38