Barycenter and affine cone

Let $G$ be an abelian group and $n\ge 0$. The barycenter of the ordered standard $n$-simplex is $b_n=(1\mathrm{/}(n+1),\dots ,1\mathrm{/}(n+1))$. An affine singular $m$-simplex in ${\mathrm{\Delta }}^n$ is determined by its ordered vertices $w_0,\dots ,w_m\in {\mathrm{\Delta }}^n$; write it as $[w_0,\dots ,w_m]$. Define its affine cone by $b_n[w_0,\dots ,w_m]=[b_n,w_0,\dots ,w_m],$ and extend by tensoring with ${\mathrm{id}}_G$ to finite affine chains. The apex is the first vertex. Repeated vertices are allowed and remain singular simplices; no quotient by degenerate simplices is being taken.

For a singular simplex $\sigma :{\mathrm{\Delta }}^n\to X$ and a supplied affine chain $\tilde{z}$ in ${\mathrm{\Delta }}^n$, define the affine cone along $\sigma$ by $b_{\sigma }\tilde{z}:={\sigma }_{\mathrm{\#}}(b_n\tilde{z})\mathrm{.}$ Here ${\sigma }_{\mathrm{\#}}$ means composition of each affine simplex with $\sigma$, with its coefficient unchanged. The lift $\tilde{z}$, not just its image chain in $X$, is part of the input. It may lie anywhere in the convex simplex, including its boundary. In the recursive notation $b_{\sigma }S(\mathrm{\partial }\sigma )$, the supplied lift is the chain $S(\mathrm{\partial }{\iota }_n)$ in $\mathrm{\partial }{\mathrm{\Delta }}^n$, where ${\iota }_n$ is the identity simplex.

With the ordinary boundary of The singular boundary operator, the orientation formulas are, for $m\ge 1$, $\mathrm{\partial }(b_{\sigma }\tilde{z})={\sigma }_{\mathrm{\#}}\tilde{z}-b_{\sigma }(\mathrm{\partial }\tilde{z}),$ and for $m=0$, $\mathrm{\partial }(b_{\sigma }\tilde{z})={\sigma }_{\mathrm{\#}}\tilde{z}-[\sigma (b_n)]\otimes \epsilon (\tilde{z}),$ where $\epsilon ({\sum }_j[w_j]\otimes g_j)={\sum }_j{g}_j$. The first formula follows by deleting the first cone vertex and then the successive base vertices with alternating signs; the second follows from $\mathrm{\partial }[b_n,w]=[w]-[b_n]$. Thus in degree zero the formula without the extra term holds only when the total coefficient is zero. In particular it applies to the subdivided boundary of a $1$-simplex, whose endpoint coefficients sum to zero. These formulas do not change the library's ordinary degree-zero boundary or its reduced-homology convention.