Exact sequence for double mapping cylinder

Definition

Let $X,Y,Z$ be topological spaces and $f:X\to Y,g:X\to Z$ be continuous maps. Let $D$ be the double mapping cylinder of $f$ and $g$. Let $i,j$ denote the inclusions of $Y$ and $Z$ in $D$. Then we have the following long exact sequence of homology:

$\dots \to H_q(X)\to H_q(Y)\oplus H_q(Z)\to H_q(D)\to H_{q-1}(X)\to \dots$

where the maps are:

$a\in H_q(X)\mapsto (H_q(f)a,-H_q(g)a)$

and:

$(b,c)\in H_q(Y)\oplus H_q(Z)\mapsto H_q(i)b+H_q(j)c$

And the third map is the usual connecting homomorphism from Mayer-Vietoris.

We can replace homology with reduced homology above.