nLab extension lemma for differential forms

Idea

The extension lemma for differential forms states that suitable differential forms on the boundary of a geometric simplex extend to (hence are the pullback along the boundary inclusion of) a differential form on the full simplex.

This statement is a crucial ingredient in the fundamental theorem of dg-algebraic rational homotopy theory. With respect to the projective model structure on connective dgc-algebras used there, it is equivalent to the condition that the simplicial dgc-algebra of differential forms on simplices is a Reedy fibrant.

Statement

Lemma

(extension lemma for PL forms) Given a PL differential form on the boundary of a simplex, it extends to (hence is the restriction of) a PL form on the full n-simplex.

(cf. Griffiths & Morgan 2013, Lemma 9.4)

Proof

Consider barycentric coordinates for the given n-simplex

$$\Delta^n \,\equiv\, \left\{ \, (x^0, \cdots, x^n) \,\in\, \mathbb{R}^{n+1} \;\Big\vert\; \textstyle{\sum}_k x^k = 1 \,, \forall_i \colon x^i \geq 0 \, \right\}$$

such that the $i$th vertex is the point $v_i \in \Delta^n$ with coordinates

$$x^k(v_i) \;=\; \delta^k_i$$

and the $i$th-face is the subset

$$\sigma_i\big(\Delta^n\big) \;=\; \big\{ \, (x^0, \cdots, x^n) \,\in\, \Delta^n \;\big\vert\; x^i = 0 \, \big\} \,.$$

With these coordinate expressions consider the functions

(1)$$\begin{array}{rcl} \mathllap{ p_i \;\colon\; } \Delta^n \setminus \{v_i\} &\longrightarrow& \sigma_i(\Delta^{n}) \\ (x^k)_{k=0}^n &\mapsto& \left( \frac{x^k}{1-x^i} \right)_{k \neq i} \,. \end{array}$$

Observe that given a polynomial form $\alpha$ on $\sigma_i(\Delta^n)$, its pullback along $p_i$ is a form on $\Delta^n \setminus \{v_i\}$ which is polynomial in the variables $\{x^k\}_{k \neq i}$ and in the variable $1/(1-x^i)$. Therefore there is a power $N_i \in \mathbb{N}$ such that

(2)$$\widehat \alpha \;\coloneqq\; (1 - x^i)^{N_i} \cdot p_i^\ast(\alpha)$$

is a differential form on $\Delta^n \setminus \{v_i\}$ such that

1. $\widehat \alpha$ extends to $v_i$ (by zero) to give a polynomial differential form on all of $\Delta^n$;

2. pulled back to the $i$th face, $\widehat \alpha$ coincides with $\alpha$ (there being the pullback of an identity map);

3. if $\alpha$ vanishes on the $j$-face in $\sigma_i(\Delta^n)$ then $\widehat{\alpha}$ vanishes on the $j$-face of $\Delta^n$ (there being a pullback of the former).

Now to complete the proof, consider a polynomial differential form $\omega$ on the boundary $\partial \Delta^n$. We need to find an extension $\widehat \omega$ to all of $\Delta^n$.

First consider the above construction (2)

$$\widehat \omega_0 \;\coloneqq\; \widehat {\omega \vert_{\sigma_0} }$$

on the restriction of $\omega$ to $\sigma_0(\Delta^n)$ and notice that the difference

$$\omega_1 \;\coloneqq\; \omega - \widehat{\omega}_0 \vert_{\partial \Delta^n}$$

vanishes on $\sigma_0(\Delta^n)$, by the second property above. Therefore consider next the above construction (2)

$$\widehat \omega_1 \;\coloneqq\; \widehat{ \omega_1 \vert_{\sigma_1} }$$

on this difference restricted to $\sigma_1$ and notice that the difference

$$\omega_2 \;\coloneqq\; \omega_1 - \widehat{\omega}_1 \vert_{\partial \Delta^n}$$

vanishes on the union of faces $\sigma_0 \cup \sigma_1$ (on $\sigma_1$ by the second property above, and on $\sigma_0$ by the third property, using that $\omega_1$ vanishes on $\sigma_0$ as just established). Proceeding in this fashion one arrives at

$$\omega \;=\; \underset{ \widehat{\omega} }{ \underbrace{ \textstyle{\sum}_k \widehat{\omega}_k } } \, \Big\vert_{\partial \Delta^n}$$

and hence the term over the brace is an extension as required.

Remark

(extension lemma for piecewise smooth differential forms) The Extension Lemma holds also for other flavors of differential forms over simplices:

In (the proof of) Griffiths & Morgan 2013, Cor. 9.9 this is observed for the case of smooth differential forms: Here the proof of Lemma applies verbatim, except that the multiplication by $(1-x^i)^{N_i}$ in (2) needs to be replaced by multiplication with any bump function $b \colon \mathbb{R} \longrightarrow \mathbb{R}$ which vanishes in a neighbourhood of $x^i = 1$ and is unity for $x^i = 0$.

This has the following further variants:

• The same argument also applies to differential forms on “extended simplices” (see this Def.)

  $$\Delta^n_{ext} \,\equiv\, \left\{ \, (x^0, \cdots, x^n) \,\in\, \mathbb{R}^{n+1} \;\Big\vert\; \textstyle{\sum}_k x^k = 1 \, \right\} \,,$$

  where the condition $\forall_i \, x^i \geq 0$ is dropped (which plays no role in the above proof).

  This is noteworthy, because it implies, with the discussion at shape via cohesive path ∞-groupoid and using the fundamental theorem of dg-algebraic rational homotopy theory, that that rational space encoded in a Sullivan model/Whitehead $L_\infty$-algebra $\mathfrak{a}$ is equivalently the shape

  $$\esh \, \Omega^1_{dR}(-;\mathfrak{a})_{flat}$$

  of the smooth set of flat $\mathfrak{a}$-valued differential forms.

Related entries

• PL de Rham theorem

References

• Phillip Griffiths, John Morgan, Chapter 9 of: Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, Birkhauser. First Edition, 1981; Second Edition, 2013. [doi:10.1007/978-1-4614-8468-4]