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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 $\delta^{-1} x$?

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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
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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 $y_A - y_B$.

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