The degreewise simplicial-to-singular homomorphisms

Let $K$ be an abstract simplicial complex whose vertex set is equipped with a total order. For each oriented $n$-simplex $\tau$ of $K$, let $[v_0<\cdots <v_n]$ be the increasing ordering of its vertices and let $\epsilon (\tau )\in \{\pm 1\}$ be the sign for which $\tau =\epsilon (\tau )[v_0,\dots ,v_n]$ in the simplicial chain group from Simplicial chain groups and the boundary operator. For each $n\ge 0$, the simplicial-to-singular homomorphism ${\mathrm{\Phi }}_n:C_n^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\to C_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\mathrm{\mid }K\mathrm{\mid };\mathbb{Z})$ is defined on an oriented simplex $\tau$ by ${\mathrm{\Phi }}_n(\tau ):=\epsilon (\tau ){\chi }_{[v_0,\dots ,v_n]},$ where ${\chi }_{[v_0,\dots ,v_n]}$ is the affine characteristic singular simplex from The affine characteristic singular simplex of an ordered simplex, and then extended linearly.

Because the increasing representative of each simplex is unique, the sign $\epsilon (\tau )$ is well defined. Thus ${\mathrm{\Phi }}_n$ is a well-defined homomorphism with no ambiguity from reordering the vertices. The next lemma proves that the family $({\mathrm{\Phi }}_n)$ commutes with the two boundaries and hence is a chain map.