= Mapping-torus homology from Mayer–Vietoris
{title2=$1-f_*$}
For a continuous self-map $f:X\to X$, the <mapping torus> has the <Wang sequence> even when $f$ is not a <homeomorphism>. Cover the time circle by a seam neighborhood and an interval away from the seam. Their inverse images retract to $X$: the seam neighborhood is a <mapping cylinder>, and the other is a product. Their intersection retracts to two copies of $X$. The <Mayer–Vietoris sequence> map on <homology (mathematics)> is, after consistent choices of signs,
$$
(a,b)\longmapsto(a+f_*b,-a-b).
$$
Invertible changes of coordinates reduce it to the direct sum of $\operatorname{id}-f_*$ and an <isomorphism>. Canceling the isomorphism summands gives the <Wang sequence>. This proof does not assume a <fiber bundle> structure for a noninvertible $f$.
Back to article page
