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 $\delta^{-1} x$?



The preimage of the boundary morphism in the Mayer-Vietoris sequence
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 $y\_A - y\_B$.


