# About the proof that $A_{PL}$ is extendable in Halperin, Thomas and Felix

I am currently reading Section 10 in Rational Homotopy Theory by Felix, Halperin and Thomas.

A rather important step after the construction of the simplicial cochain algebra $A = A_{PL}$ is proving they are extendable, meaning that for any (not necessarily strict) subset $I$ of $\{0,\ldots,n\}$ and simplices $\Phi_i$ for which $\partial_i\Phi_j = \partial_{j-1}\Phi_i$, $i<j$ in $I$, there is $\Phi$ such that $\partial_i \Phi = \Phi_i$.

This is claim $(iii)$ in Lemma 10.7, page 123 of the first edition.

Concretely, $A_n$ is the free commutative DG-algebra generated by $t_0,\ldots,t_n$ in degree $0$, by $y_0,\ldots,y_n$ in degree $1$ and with $dt_i = y_i$, modulo the relations that $\sum t_i = 1$ and $\sum y_i=0$, face/coface maps given by thinking about the $t_i$ as the barycentric coordinates of the $n$-simplex. Equivalently, it is generated by $t_1,\ldots,t_n$ and $y_1,\ldots,y_n$ subject to commutativity only and $t_i$ covers $y_i$.

I find the proof really obscure; although with a bit of work I can follow it, I cannot get anything out of it. I was wondering if someone could shed some light on the proof, or else point to other proofs (if there are any). I admit the whole book is rather terse at moments, but in general, independent of this, I can gather some intuition of what is going on, but this particular bit is eluding me.

In few words: Let $$p\colon\Delta^n\to \Delta^{n-1} = \{t_0 = 0\}$$ be a stereographic projection from the vertex $$t_0 = 1$$, and let $$\alpha \in \Omega^*(\Delta^{n-1})$$ be a polynomial form on the face $$t_0 = 0$$. Then $$p^*(\alpha)$$ is polynomial from from generators $$t_0, \dots, t_{n-1}, \frac{1}{1-t_n}, dt_1, \dots, dt_n$$. Therefore, $$(1-t_n)^Np^*(\alpha)$$ is actually polynomial form on $$\Delta^n$$.
Let $$\varphi \in \Omega^*(\partial\Delta[n])$$. We now build forms $$\psi_j$$, s.t. $$\varphi - \sum\limits_{0}^k \psi_i$$ vanishes on the union $$\sigma_0\cup\dots\cup\sigma_k$$, where $$\sigma_i = d^i[n-1]$$ is $$(n-i)$$-th face. Namely, let $$\psi_0$$ be an extension of $$\varphi|_{\sigma_0}$$ to the full simplex $$\Delta[n]$$. Now, let $$\psi_1$$ be an extension of $$\varphi_1 = (\varphi - \psi_0)|_{\sigma_1}$$. Then $$\varphi - \psi_0 - \psi_1$$ vanishes on $$\sigma_0\cup\sigma_1$$. Now, we continue inductively. Hence, we get $$\varphi = \sum\limits_{i = 0}^n\psi_i|_{\partial\Delta[n]}$$, proving that $$\Omega^*$$ is extandable.