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

Preliminaries

Write

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

for the simplicial object in dgc-algebras given by polynomial differential forms on simplices over the real numbers (see also at fundamental theorem of dgc-algebraic rational homotopy theory – change of scalars).

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

(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:

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

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:

• 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).

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

• 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)