nLab PL de Rham theorem




The PL de Rham theorem is the variant of the de Rham theorem with the smooth de Rham complex replaced by the PL de Rham complex.



(PL de Rham theorem)

Let kk be a field of characteristic zero (such as the rational numbers, real numbers or complex numbers).

Then the evident operation of integration of differential forms over simplices induces a quasi-isomorphism between the PL de Rham complex with coefficients in kk and cochain complex for singular cohomology with coefficients in kk

Ω PLdR (X)C (X;k) \Omega^\bullet_{PLdR}(X) \underoverset{}{\simeq}{\longrightarrow} C^\bullet(X; k)

and hence an isomorphism from PL de Rham cohomology to ordinary cohomology with coefficients in kk (such as rational cohomology, real cohomology, complex cohomology):

H PLdR (X)H (X;k) H^\bullet_{PLdR}(X) \underoverset{}{\simeq}{\longrightarrow} H^\bullet(X; k)

(for XX any topological space).

(Bousfield-Gugenheim 76, Theorem 2.2)


Besides the Poincaré lemma for piecewise polynomial forms, the proof uses:


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

(e.g. Griffiths & Morgan 2013, Lemma 9.6)


Consider barycentric coordinates for the given n-simplex

Δ n{(x 0,,x n) n+1| kx k=1, ix i0} \Delta^n \,\equiv\, \left\{ \, (x^0, \cdots, x^n) \,\in\, \mathbb{R}^{n+1} \;\Big\vert\; \textstyle{\sum}_k x^k = 1 \,, \forall_i \, x^i \geq 0 \, \right\}

such that the iith vertex is the point v iΔ nv_i \in \Delta^n with coordinates

x n(v i)=δ i n x^n(v_i) \;=\; \delta^n_i

and the iith-face is the subset

σ i(Δ n)={(x 0,,x n)Δ n|x i=0}. \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)p i:Δ n{v i} σ i(Δ n) (x k) k (x k1x i) ki. \begin{array}{rcl} \mathllap{ p_i \;\colon\; } \Delta^n \setminus \{v_i\} &\longrightarrow& \sigma_i(\Delta^{n}) \\ (x^k)_{k} &\mapsto& \left( \frac{x^k}{1-x^i} \right)_{k \neq i} \,. \end{array}

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

(2)α^(1x i) N ip i *(α) \widehat \alpha \;\coloneqq\; (1 - x^i)^{N_i} \cdot p_i^\ast(\alpha)

is a differential form on Δ n{v i}\Delta^n \setminus \{v_i\} such that

  1. α^\widehat \alpha extends to v iv_i (by zero) to give a polynomial differential form on all of Δ n\Delta^n;

  2. pulled back to the iith face, α^\widehat \alpha coincides with α\alpha (there being the pullback of an identity map);

  3. if α\alpha vanishes on the jj-face in σ(Δ n)\sigma(\Delta^n) then α^\widehat{\alpha} vanishes on the jj-face of Δ n\Delta^n (there being a pullback of the former).

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

First consider the above construction (2)

ω^ 0ω| σ 0^ \widehat \omega_0 \;\coloneqq\; \widehat {\omega \vert_{\sigma_0} }

on the restriction of ω\omega to σ 0(Δ n)\sigma_0(\Delta^n) and notice that the difference

ω 1ωω^ 0| Δ n \omega_1 \;\coloneqq\; \omega - \widehat{\omega}_0 \vert_{\partial \Delta^n}

vanishes on σ 0\sigma_0 and coincides with ω\omega on all other faces. Therefore consider next the above construction (2)

ω^ 2ω 1| σ 1^ \widehat \omega_2 \;\coloneqq\; \widehat{ \omega_1 \vert_{\sigma_1} }

on this difference restricted to σ 1\sigma_1 and notice that the difference

ω 2ω 1ω^ 1| Δ n \omega_2 \;\coloneqq\; \omega_1 - \widehat{\omega}_1 \vert_{\partial \Delta^n}

vanishes on the union of faces σ 0σ 1\sigma_0 \cup \sigma_1 and coincides with ω\omega on all remaining faces. Proceeding in this fashion one arrives at

ω= kω^ kω^| Δ n \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.


(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 (1x i) N i(1-x^i)^{N_i} in (2) needs to be replaced by multiplication with any bump function which vanishes in a neighbourhood of x i=1x^i = 1 and is unity for x i=0x^i = 0.

Moreover, we may observe that this same argument then also applies to differential forms on “extended simplices” (see this Def.)

Δ ext n{(x 0,,x n) n+1| kx k=1}, \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 ix i0\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 L_\infty -algebra 𝔞\mathfrak{a} is equivalently the shape

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

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


Including the variant of piecewise smooth forms:

Last revised on February 3, 2024 at 18:13:37. See the history of this page for a list of all contributions to it.