Mapping telescope

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