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

Contents

Context

Rational homotopy theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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)Ω polydR (Δ ):Δ opdgcAlgebras 0 \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 theorychange of scalars).

Definition

(PL de Rham complex)

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

(2)Ω PLdR (S)sSet(S,Ω polydR (Δ ))[n]Δ opS nΩ polydR (Δ n). \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 XX \in Top a topological space its PL de Rham complex is the PL de Rham complex as in (2) of its singular simplicial complex:

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

Write

(3)Ω dR ():Δ opdgcAlgebras 0 \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 SS \in sSet a simplicial set, its PS de Rham complex (“piecewise smooth”) is the hom-object of simplicial objects from SS to Ω dR (Δ )\Omega^\bullet_{dR}(\Delta^\bullet) (3), hence is the following end in dgcAlgebras:

(4)Ω PSdR (S)sSet(S,Ω dR (Δ ))[n]Δ opS nΩ dR (Δ n). \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)Ω PLdR ()AAi polyAAΩ PSdR () \Omega_{PLdR}^\bullet(-) \overset{ \phantom{AA} i_{poly} \phantom{AA} }{\hookrightarrow} \Omega_{PSdR}^\bullet(-)

For XX a smooth manifold, and S(X)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 XX 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) XX and the smooth de Rham complex of XX:

Ω PLdR (S(X)) Ω dR (X) i * i poly p * Ω PLdR (X)=Ω PLdR (Sing(X)) Ω PSdR (S(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)S(X) is the simplicial complex corresponding to any smooth triangulation of XX.

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)Sing(X)S(X) \hookrightarrow Sing(X) is a weak homotopy equivalence and since Ω PLdR \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.