nLab PL de Rham complex

Idea

The concept of the PL de Rham complex (Bousfield-Gugenheim 76, Sullivan 77) is a variant of that of the de Rham complex for smooth manifolds which applies to general topological spaces and simplicial sets.

The terminology “PL” for “piecewise linear” seems to have been tacitly introduced in Bousfield-Gugenheim 76. Beware that despite this commonly adopted terminology (e.g. Griffith-Morgan 13), the PL de Rham complex consist of piecewise polynomial differential forms, but polynomial with respect to a piecewise linear structure on the domain space/complex.

In analogy to the de Rham theorem for smooth manifolds, the fundamental theorem of dg-algebraic rational homotopy theory shows that the PL de Rham complex computes the rational cohomology (or real cohomology, complex cohomology) of the given topological space/simplicial sets.

Applied to a topological space that happens to carry the structure of a smooth manifold, the PL de Rham complex is connected by a zig-zag of quasi-isomorphisms to the smooth de Rham complex, hence both are isomorphic in the homotopy category of the model structure on connective dgc-algebras.

Definition

Let $k \in \{\mathbb{Q}, \mathbb{R}, \mathbb{C}\}$.

Write

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

for the simplicial object in dgc-algebras given by polynomial differential forms on simplices.

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:

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

(Bousfield-Gugenheim 76, Sec. 2, p. 7)

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:

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

The cochain cohomology of the PL de Rham complex is PL de Rham cohomology

(3)$$H^\bullet_{PLdR}(-) \;\coloneqq\; H \Omega^\bullet_{PLdR}(-) \,.$$

Properties

Relation to simplicial sets

Proposition

(Quillen adjunction between simplicial sets and connective dgc-algebras)

The PL de Rham complex-construction (Def. ) is the left adjoint in a Quillen adjunction between

• the opposite of the projective model structure on connective dgc-algebras

• the classical model structure on simplicial sets

$$\big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} SimplicialSets_{Qu}$$

Proof

That the PL de Rham complex functor preserves cofibrations, hence sends injections of simplicial sets to surjections of dgc-algebras, is immediate from its construction.

That its right adjoint preserves fibrations, hence sends cofibrations of dgc-algebras to Kan fibrations, is the statement of Bousfield-Gugenheim 76, Lemma 8.2.

Relation to rational cohomology

Proposition

(PL de Rham theorem)

Let $k$ 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 $k$ (Def. }) and cochain complex for singular cohomology with coefficients in $k$

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

and hence an isomorphism from PL de Rham cohomology (3) to ordinary cohomology with coefficients in $k$ (such as rational cohomology, real cohomology, complex cohomology):

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

(for $X$ any topological space).

(Bousfield-Gugenheim 76, Theorem 2.2)

Relation to rational homotopy type

Definition

(nilpotent and finite rational homotopy types)

Write

(4)$$Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)$$

for the full subcategory of the classical homotopy category (homotopy category of the classical model structure on simplicial sets) on those homotopy types $X$ which are

• connected: $\pi_0(X) = \ast$

• nilpotent: $\pi_1(X)$ is a nilpotent group

• rational finite type: $dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty$ for all $n \in \mathbb{N}$.

and

(5)$$Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)$$

for the futher full subcategory on those homotopy types that are already rational.

Similarly, write

(6)$$Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)_{fin}^{\geq 1} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}} \big)$$

for the full subcategory of the homotopy category of the projective model structure on connective dgc-algebras on those dgc-algebras $A$ which are

• connected: $H^0(A) \simeq \mathbb{Q}$

• finite type: $dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty$ for all $n \in \mathbb{N}$.

(Bousfield-Gugenheim 76, 9.2)

Proposition

(fundamental theorem of dg-algebraic rational homotopy theory)

The derived adjunction

$$Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right) \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot} Ho \big( HoSimplicialSets_{Qu} \big)$$

of the Quillen adjunction between simplicial sets and connective dgc-algebras (whose left adjoint is the PL de Rham complex-functor) has the following properties:

• on connected, nilpotent rationally finite homotopy types $X$ (4) the derived adjunction unit is rationalization

  $$\array{ Ho \big( SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} & \overset{ }{\longrightarrow} & Ho \big( SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \\ X &\mapsto& \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X) }$$
  $$X \underoverset {\eta_X^{der}} {rationalization} {\longrightarrow} \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)$$

• on the full subcategories of nilpotent and finite rational homotopy types from Def. it restricts to an equivalence of categories:

  $$Ho \left( \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( HoSimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}$$

(Bousfield-Gugenheim 76, Theorems 9.4 & 11.2)

Relation to smooth de Rham complex

Write

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

for the simplicial object in dgc-algebras 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 $\Omega^\bullet_{dR}(\Delta^\bullet)$ (7), hence is the following end in dgcAlgebras:

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

This receives an evident inclusion from the PL de Rham complex (8):

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

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

• there is a homeomorphism of topological spaces

  $$S(X) \overset{p}{\longrightarrow} X$$

  which restricts to a diffeomorphism onto its image in the interior of any simplex

• there is a weak homotopy equivalence of simplicial sets

  $$S(X) \overset{ \phantom{AA} i \phantom{AA} }{\hookrightarrow} Sing(X) = Sing\big( \left\vert S(X)\right\vert \big)$$

  into the singular simplicial set of $X$ (this being the adjunction unit of the $\left\vert - \right\vert \dashv Sing$ Quillen equivalence between the classical model structure on simplicial sets and the classical model structure on topological spaces, and in fact equivalently the derived adjunction unit, since $\left\vert -\right\vert$ preserves all weak equivalences, and $Sing$ those between CW-complexes, by Ken Brown's lemma).

Definition

(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.

Related concepts

• de Rham complex

• equivariant PL de Rham complex

• rational homotopy theory

• fundamental theorem of dg-algebraic rational homotopy theory

References

• Dennis Sullivan, Infinitesimal computations in topology, Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam:PMIHES_1977__47__269_0)

• Aldridge Bousfield, Victor Gugenheim, On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)

• Phillip Griffiths, John Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (2013) (doi:10.1007/978-1-4614-8468-4)