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.

enter image description here

enter image description here

up vote 3 down vote accepted

Don't know is this question still actual, but you can find an answer below.

First of all, you can find more geometric proof in Griffiths, Morgan -- Rational Homotopy Theory and Differential Forms.

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.

New contributor
Semyon Abramyan is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.