# nLab PL de Rham complex of smooth manifold is equivalent to de Rham complex

Contents

and

## Sullivan models

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Preliminaries

Write

(1)$\Omega^\bullet_{polydR} (\Delta^\bullet) \;\colon\; \Delta^\op \longrightarrow dgcAlgebras^{\geq 0}_{\mathbb{R}}$
###### Definition

(PL de Rham complex)

The for $S \in$ sSet a simplicial set, its PL de Rham complex is the hom-object of simplicial objects from $S$ to $\Omega^\bullet_{polyDR}$ (1), hence is the following end in dgcAlgebras, here over the real numbers:

(2)$\Omega^\bullet_{PLdR}(S) \;\coloneqq\; sSet \big( S,\, \Omega^\bullet_{polydR}(\Delta^\bullet) \big) \;\coloneqq\; \underset{ [n] \in \Delta^{op} }{\int} \underset{ S_n }{\oplus} \Omega^\bullet_{polydR} \big( \Delta^n \big) \,.$

For $X \in$ Top a topological space its PL de Rham complex is the PL de Rham complex as in (2) of its singular simplicial complex:

$\Omega^\bullet_{PLdR}(X) \;\coloneqq\; \Omega^\bullet_{PLdR} \big( Sing(X) \big) \,.$

Write

(3)$\Omega^\bullet_{dR} (-) \;\colon\; \Delta^\op \longrightarrow dgcAlgebras^{\geq 0}_{\mathbb{R}}$

for the simplicial object in dgc-algebras (over the real numbers) given by smooth differential forms on simplices.

###### Definition

(PS de Rham complex)

The for $S \in$ sSet a simplicial set, its PS de Rham complex (“piecewise smooth”) is the hom-object of simplicial objects from $S$ to $\Omega^\bullet_{dR}(\Delta^\bullet)$ (3), hence is the following end in dgcAlgebras:

(4)$\Omega^\bullet_{PSdR}(S) \;\coloneqq\; sSet \big( S,\, \Omega^\bullet_{dR}(\Delta^\bullet) \big) \;\coloneqq\; \underset{ [n] \in \Delta^{op} }{\int} \underset{ S_n }{\oplus} \Omega^\bullet_{dR} \big( \Delta^n \big) \,.$

This receives an evident inclusion from the PL de Rham complex (4):

(5)$\Omega_{PLdR}^\bullet(-) \overset{ \phantom{AA} i_{poly} \phantom{AA} }{\hookrightarrow} \Omega_{PSdR}^\bullet(-)$

For $X$ a smooth manifold, and $S(X)$ the simplicial complex given by any smooth triangulation, notice that:

## Statement

###### Proposition

(PL de Rham complex of smooth manifold is equivalent to de Rham complex)

Let $X$ be a smooth manifold.

We have the following zig-zag of dgc-algebra quasi-isomorphisms between the PL de Rham complex of (the topological space underlying) $X$ and the smooth de Rham complex of $X$:

$\array{ && \Omega^\bullet_{PLdR} \big( S(X) \big) && && \Omega^\bullet_{dR}(X) \\ & {}^{ \mathllap{ i^\ast } } \nearrow & & \searrow^{ \mathrlap{ i_{poly} } } & & {}^{ \mathllap{ p^\ast } } \swarrow \\ \mathllap{ \Omega^\bullet_{PLdR}(X) \;=\; } \Omega^\bullet_{PLdR} \big( Sing(X) \big) && && \Omega^\bullet_{PSdR} \big( S(X) \big) }$

Here $S(X)$ is the simplicial complex corresponding to any smooth triangulation of $X$.

###### Proof

For the two morphisms on the right this is Griffith-Morgan 13, Cor. 9.9.

For the morphism on the left this follows since $S(X) \hookrightarrow Sing(X)$ is a weak homotopy equivalence and since $\Omega^\bullet_{PLdR}$, being a left Quillen functor preserves weak equivalences between cofibrant objects (where every simplicial set being cofibrant), by Ken Brown's lemma.

## References

Last revised on July 7, 2021 at 12:15:15. See the history of this page for a list of all contributions to it.