Two questions on axiomatic homology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:32:39Z http://mathoverflow.net/feeds/question/109738 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109738/two-questions-on-axiomatic-homology Two questions on axiomatic homology FJH 2012-10-15T17:46:24Z 2012-10-15T17:46:24Z <p>1) Given the Eilenberg-Steenrod axioms, there are several Mayer-Vietoris type sequences that can be deduced. The most general form seems to be</p> <p>$$\rightarrow H_n(X \cap Y, A \cap B) \rightarrow H_n(X,A) \oplus H_n(Y,B) \rightarrow H_n(X \cup Y, A \cup B) \rightarrow H_{n-1}(X \cap Y, A \cap B) \rightarrow$$</p> <p>where $A,B,X,Y$ are (say) open with $A \subseteq X, B \subseteq Y$.</p> <p>Question: How can one define the connecting map for this sequence? </p> <p>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.</p> <p>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. </p> <p>Question: Is this also true for general $Y$?</p> <p>(Of course it is if $H$ sends weak equivalences to isomorphisms)</p>