Exact sequence for double mapping cylinder

This article defines a long exact sequence of homology groups, for topological spaces or pairs of topological spaces

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:

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

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.