Oriented simplicial subdivision operator

On oriented integral chains define the graded subdivision operator $S:C_n(K)\to C_n(\mathrm{sd}K)$ by sending an oriented simplex to the sum of its top-dimensional chain simplices with the orientation inherited through the barycentric realization. The triangulation in Barycentric face chains triangulate a geometric simplex makes these pieces nondegenerate. The source and target groups use Simplicial chain groups and the boundary operator, and target vertices are faces as in Barycentric subdivision of an abstract simplicial complex.

Equivalently, using Augmentation and reduced simplicial homology, augment with $C_{-1}=\mathbb{Z}$, $\mathrm{\partial }[v]=1$, set $S_{-1}=1$, and recursively set $S(s)=c_{\sigma }S(\mathrm{\partial }s),c_{\sigma }[t_0,\dots ,t_j]=[\sigma ,t_0,\dots ,t_j],c_{\sigma }(1)=[\sigma ],$ where $\sigma$ is the underlying face of $s$. Every face label occurring in $S(\mathrm{\partial }s)$ is a proper face of $\sigma$, so coning is defined. The boundary of a geometric oriented cone with its apex first induces the given orientation on its opposite face: this follows from the positive coefficient of that face in the alternating boundary. Coning the oriented boundary triangulation therefore gives precisely the inherited orientations of the pieces. Reversing the original orientation changes every summand's sign, so this defines the graded operator on oriented chains. In degree zero $S[v]=[\{v\}]$. Boundary compatibility is a separate result; no chain-map property is assumed here.

2.1, pp.121–122, recursive subdivision.