Mapping cylinder

The mapping cylinder of $f:A\to X$ isIt deformation retracts onto the copy of $X$.

The topological mapping cone of $f:X\to Y$ is $Y{\cup }_{f}CX$, where $CX$ is the cone on $X$ and its base is identified with $Y$ through $f$. For nonempty $X$, the cone vertex supplies a distinguished basepoint. This construction is related to, but distinct from, the mapping cone of a chain map.

For a cellular map between nonempty CW complexes, the reduced cellular chain complex of its topological mapping cone has differentialHere $C_{-1}(X)=0$ and the degree-zero group identifies a vertex $y$ with the reduced chain $y-v$, where $v$ is the cone vertex. Thus this is the mapping cone of $f_{\#}$, in the displayed ordering of the summands.

For nonempty $X$, the topological mapping cone has this long exact sequence in homology, with ordinary homology for $X,Y$ and reduced homology for $C_f$. In degree zero, the augmentation of $X$ is accounted for by the cone vertex. The sequence follows from the Mayer–Vietoris theorem applied to neighborhoods of the cone and $Y$.

The mapping telescope of $X_0\stackrel{f_0}{\to }{X}_1\stackrel{f_1}{\to }\cdots$ glues the mapping cylinders of the $f_i$ end to end. Its homology is naturally the direct limit ${\underrightarrow{\lim }}_i{H}_{\ast }(X_i)$.

List the primes as $p_0,p_1,\dots$ and form the mapping telescope of degree-$p_i$ maps $S^n\to S^n$. Its reduced homology vanishes outside degree $n$, while