MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose $A$ and $B$ are contractible pointed open subspaces of $X$. The Mayer-Vietoris sequence implies that the boundary morphism $\delta: H_n(X)\to H_{n-1}(A\cap B)$ is an isomorphism.

I wonder, if $x$ is a $n-1$-cycle of $A\cap B$, is there an explicit algorithm to write down its $n$-cycle preimage $\delta^{-1} x$?

share|cite|improve this question
Yes, the $n$-cycle is the union of the two cones on $x$. The first cone you get via the deformation-retraction in $A$, the 2nd cone you get by the deformation-retraction in $B$. – Ryan Budney May 31 '11 at 3:13
up vote 11 down vote accepted

Take a contraction of $x$ in $A$. That gives an $n$-chain $y_A$ living in $A$, whose boundary is $x$. Similarly, a contraction of $x$ in $B$ gives an $n$-chain $y_B$, again with boundary equal to $x$. The desired preimage is $y_A - y_B$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.