Mayer–Vietoris connecting class

Fix an abelian group $G$, an open cover $X=U\cup V$, and $n\ge 1$. For a cover-small $n$-cycle $z=u+v$ with $u\in C_n(U;G)$ and $v\in C_n(V;G)$, define $\delta [z]:=[\mathrm{\partial }u]=[-\mathrm{\partial }v]\in H_{n-1}(U\cap V;G)\mathrm{.}$ This is the connecting-class convention compatible with $i(c)=(c,-c)$.

Indeed, $\mathrm{\partial }z=0$ implies $\mathrm{\partial }u=-\mathrm{\partial }v$. As a chain lying in both subcomplexes, this is a chain on $U\cap V$, and ${\mathrm{\partial }}^{2}u=0$ makes it a cycle. For $n=1$ this uses the ordinary zero boundary on $0$-chains; if the overlap is empty the chain is zero.

The well-definedness obligation is discharged by Well-definedness of the Mayer–Vietoris connector ↗. Explicitly, replacing $(u,v)$ by another decomposition changes $u$ by an overlap chain and hence $\mathrm{\partial }u$ by an overlap boundary. If $z^{\mathrm{\prime }}-z=\mathrm{\partial }(a+b)$ in the small complex, where $a$ lies in $U$ and $b$ in $V$, then $u^{\mathrm{\prime }}-u-\mathrm{\partial }a$ is an overlap chain and its boundary is $\mathrm{\partial }{u}^{\mathrm{\prime }}-\mathrm{\partial }u$. The cover-small homology identification used in Mayer–Vietoris sequence in singular homology therefore makes this a class depending only on $[z]\in H_n(X;G)$. Lifting $z$ to $(u,v)$ gives boundary $(\mathrm{\partial }u,-\mathrm{\partial }u)=i(\mathrm{\partial }u)$, so the sign agrees with that sequence.