An affine cone homotopy from the diagonal to the front-back shuffle

Affine simplices and their faces stay in $Q_n$ by convexity. Postcomposition by an affine map commutes with the alternating boundary, term by term. The face identities give ${\mathrm{\partial }}^2=0$: deleting vertices in positions $i<j$ occurs once with sign $(-1{)}^{i+j-1}$ and once with sign $(-1{)}^{i+j}$. By [F2], [F3] and [F4], both $a_n$ and $b_n$ have boundary equal to the alternating sum of their corresponding face-model chains. Explicitly, with $f_i={\delta }_i\times {\delta }_i$, $\mathrm{\partial }(a_n-b_n)={\sum }_i(-1{)}^i(f_i{)}_{\mathrm{\#}}(a_{n-1}-b_{n-1})\mathrm{.}$ The shuffle formula commutes with the $f_i$ since each of its affine vertex lists is postcomposed unchanged.

Fix the vertex $w=(v_0,v_0)$ of $Q_n$ and define its affine cone $c_w[w_0,\dots ,w_j]=[w,w_0,\dots ,w_j]$, extended linearly. Deleting the first vertex gives the input simplex. Deleting vertex $i+1$ gives minus the cone on its $i$th face. Thus for every positive-degree chain $z$, $\mathrm{\partial }{c}_{w}z=z-c_w\mathrm{\partial }z\mathrm{.}$ For degree-zero chains the formula is $\mathrm{\partial }{c}_{w}z=z-ϵ(z)[w]$, where $ϵ$ is the sum of coefficients. All these chains are finite and affine, including cones on degenerate simplices.

Set $h_0=0$, since the sole shuffle gives $b_0=a_0$. Suppose the chains through $h_{n-1}$ have been defined with the displayed identity, and put $z_n=a_n-b_n-{\sum }_i(-1{)}^i(f_i{)}_{\mathrm{\#}}{h}_{n-1}\mathrm{.}$ For $n=1$, step 1.1 gives $\mathrm{\partial }(a_1-b_1)={\sum }_i(-1{)}^i(f_i{)}_{\mathrm{\#}}(a_0-b_0)=0$, so $\mathrm{\partial }{z}_1=0$. For $n\ge 2$, substitute the already proved boundary of $h_{n-1}$ into $\mathrm{\partial }{z}_n$. The terms involving $a_{n-1}-b_{n-1}$ cancel by step 1.1. The remaining double sum is zero: each omitted pair of vertices occurs in the two orders with opposite alternating signs, and both coordinate factors use the same composite face map. Therefore $\mathrm{\partial }{z}_n=0$ in all cases $n\ge 1$.

Define $h_n=c_{(v_0,v_0)}{z}_n$. Since $z_n$ has positive degree and is a cycle, step 2.1 gives $\mathrm{\partial }{h}_n=z_n$, exactly the claimed equation. This recursion specifies a unique chosen formula, not a unique possible filling: the cone vertex, signs and face chains have all been fixed. At each degree only finitely many faces and finite chains occur. Induction therefore defines every $h_n$ without making selections from families of possible fillers.

A smooth form supplied on a neighbourhood of $Q_n$ in its affine span pulls back by any of these affine simplex maps to a locally extendible form on the standard simplex. Integration in [F5] is linear, so the chain equation can be integrated term by term. Independence from an ambient extension follows because two extensions agree, with all their derivatives, on the interior of $Q_n$ and by continuity on its closure. The affine span has dimension zero only when $n=0$, where integration is evaluation and the equation is already zero. Applying the simplex Stokes clause of [F6] to each affine summand is legitimate regardless of its rank; no Stokes theorem on a manifold with corners has been assumed.

The zero input chain and empty finite sum give zero throughout. At $n=0$ there is no face term and $h_0=0$; the first positive degree uses the separate cycle verification in step 3.1. Repeated vertices and endpoint faces were retained in the face and cone identities. Every coefficient and map in the recursion is specified, so neither countable nor arbitrary choice is used.