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differential forms on simplices

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Homotopy theory

Contents

Idea

There are various simplicial dg-algebras that assign to the standard nn-simplex a kind of de Rham algebra on Δ n\Delta^n.

By the discussion at differential forms on presheaves, each such extends to a notion of differential forms on simplicial sets.

Definition

Smooth differential forms

Definition

(smooth nn-simplex)

For nn \in \mathbb{N} the smooth n-simplex Δ smth n\Delta^n_{smth} is the smooth manifold with boundary and corners defined, up to isomorphism, as the following locus inside the Cartesian space n+1\mathbb{R}^{n+1}:

Δ smth n{(x 0,x 1,,x n) n+1|0x i1andi=0nx i=0} n+1. \Delta^n_{smth} \;\coloneqq\; \left\{ (x_0, x_1, \cdots, x_n) \in \mathbb{R}^{n+1} \;\vert\; 0 \leq x_i \leq 1 \;\text{and}\; \underoverset{i = 0}{n}{\sum} x_i \; = 0 \right\} \hookrightarrow \mathbb{R}^{n+1} \,.

For 0in0 \leq i \leq n the function

x i:Δ smth n x_i \;\colon\; \Delta^n_{smth} \to \mathbb{R}

which picks the iith component in the above definition is called the iith barycentric coordinate function.

For

f:[n 1][n 2] f \;\colon\; [n_1] \longrightarrow [n_2]

a morphism of finite non-empty linear orders [n]{0<1<<n}[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}, let

Δ smth(f):Δ smth n 1Δ smth n 2 \Delta_{smth}(f) \;\colon\; \Delta^{n_1}_{smth} \longrightarrow \Delta^{n_2}_{smth}

be the smooth function defined by x ix f(i)x_i \mapsto x_{f(i)}.

Definition

(smooth differential forms on the smooth nn-simplex)

For kk \in \mathbb{N} then a smooth differential k-form on the smooth nn-simplex (def. 1) is a smooth differential form in the sense of smooth manifolds with boundary and corners. Explicitly this means the following.

Let

F n{(x 0,x 1,,x n) n+1|i=0nx i=0} n+1 F^n \;\coloneqq\; \left\{ (x_0, x_1, \cdots, x_n) \in \mathbb{R}^{n+1} \;\vert\; \underoverset{i = 0}{n}{\sum} x_i \; = 0 \right\} \hookrightarrow \mathbb{R}^{n+1}

be the affine plane in n+1\mathbb{R}^{n+1} that contains Δ smth n\Delta^n_{smth} in its defining inclusion from def. 1. This is a smooth manifold diffeomorphic to the Cartesian space n\mathbb{R}^{n}.

A smooth differential form on Δ smth n\Delta^n_{smth} of degree k$ is a collection of linear functions

kT xF n \wedge^k T_x F^n \longrightarrow \mathbb{R}

out of the kk-fold skew-symmetric tensor power of the tangent space of F nF^n at some point xx to the real numbers, for all xΔ smth nx \in \Delta^n_{smth} such that this extends to a smooth differential kk-form on F nF^n.

Write Ω (Δ smth n)\Omega^\bullet(\Delta^n_{smth}) for the graded real vector space defined this way. By definition there is then a canonical linear map

Ω (F n)Ω (Δ smth n) \Omega^\bullet(F^n) \longrightarrow \Omega^\bullet(\Delta^n_{smth})

from the de Rham complex of F nF^n and there is a unique structure of a differential graded-commutative algebra on Ω (Δ smth n)\Omega^\bullet(\Delta^n_{smth}) that makes is a homomorphism of dg-algebras form the de Rham algebra of F nF^n. This is the de Rham algebra of smooth differential forms on the smooth nn-simplex.

For f:[n 1][n 2]f \colon [n_1] \to [n_2] a homomorphism of finite non-empty linear orders with Δ smth(f):Δ smth n 1Δ smth n 2\Delta_{smth}(f) \colon \Delta^{n_1}_{smth} \to \Delta^{n_2}_{smth} the corresponding smooth function according to def. 1, there is the induced homomorphism of differential graded-commutative algebras

(Δ smth(f)) *:Ω (Δ smth n 2)Ω (Δ smth n 1) (\Delta_{smth}(f))^\ast \;\colon\; \Omega^\bullet(\Delta^{n_2}_{smth}) \longrightarrow \Omega^\bullet(\Delta^{n_1}_{smth})

induced from the usual pullback of differential forms on F nF^n. This makes smooth differential forms on smooth simplices be a simplicial object in differential graded-commutative algebras:

Ω (Δ smth ()):Δ opdgcAlg . \Omega^\bullet(\Delta^{(-)}_{smth}) \;\colon\; \Delta^{op} \longrightarrow dgcAlg_{\mathbb{R}} \,.

The standard proof of the Poincaré lemma applies to show that

H (Ω (Δ smth n)). H^\bullet(\Omega^\bullet(\Delta^n_{smth})) \simeq \mathbb{R} \,.

Each element of Ω poly p(Δ n)\Omega^p_{poly}(\Delta^n) may be uniquely written

Φ= 1i 1<<i pnΦ i 1i pdb i 1db i p,\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},

where b jb_j is as above the j thj^{th} barycentric coordinate function and each Φ i 1i p\Phi_{i_1\ldots i_p} is a C C^\infty-function on Δ n\mathbf{\Delta}^n.

With this representation the multiplication and differential are given by the usual formulae. The multiplication is defined by ΦΨ\Phi \wedge \Psi and extends linearly the product

(db i 1db i p)(db j 1db j q)=(db i 1db i pdb j 1db j q)(d b_{i_1}\wedge\ldots d b_{i_p})\wedge (d b_{j_1}\wedge\ldots d b_{j_q}) = (d b_{i_1}\wedge\ldots d b_{i_p}\wedge d b_{j_1}\wedge\ldots d b_{j_q})

on the generating forms. Now if ff is a differentiable function

df= i=1 nfx idx i,d f = \sum_{i=1}^{n} \frac{\partial f}{\partial x^i}d x_i,

so if

Φ= 1i 1<<i pnΦ i 1i pdb i 1db i p,\Phi =\sum_{1\leq i_1\lt \ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},

then

dΦ= 1i 1<<i pndΦ i 1i pdb i 1db i p,d\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n} d\Phi_{i_1\ldots i_p} \wedge d b_{i_1} \wedge \ldots d b_{i_p},

Polynomial differential forms

Definition

For nn \in \mathbb{N} write

Ω poly (Δ n)Sym t 0,,t n,dt 0,,dt n/(t i1,dt i) \Omega_{poly}^{\bullet}(\Delta^n) \;\coloneqq\; Sym^\bullet_{\mathbb{Q}} \langle t_0, \cdots, t_n, d t_0, \cdots, d t_n\rangle/\left(\sum t_i -1, \sum d t_i \right)

for the quotient of the \mathbb{Z}-graded symmetric algebra over the rational numbers on n+1n+1 generators t it_i in degree 0 and n+1n+1 generators dt id t_i of degree 1.

In particular in degree 0 this are called the polynomial functions

Ω poly 0(Δ n)=[t 0,t 1,t n]/(it i=0) \Omega^0_{poly}(\Delta^n) \;=\; \mathbb{Q}[t_0, t_1, \cdots t_n]/\left( \underset{i}{\sum} t_i = 0 \right)

due to the canonical inclusion

Ω poly 0(Δ n)C (Δ smth n) \Omega^0_{poly}(\Delta^n) \hookrightarrow C^\infty(\Delta^n_{smth})

into the smooth functions on the nn-simplex according to def. 2, obtained by regarding the generator t it_i as the iith barycentric coordinate function.

Observe that the tensor product of the polynomial differential forms over these polynomial functions with the smooth functions on the nn-simplex, is canonically isomorphic to the space Ω (Δ smth n)\Omega^\bullet(\Delta^n_{smth}) of smooth differential forms, according to def. 2:

Ω (Δ smth n)C (Δ smth n) Ω poly 0(Δ n)Ω poly (Δ n) \Omega^\bullet(\Delta^n_{smth}) \simeq C^\infty(\Delta^n_{smth}) \otimes_{\Omega^0_{poly}(\Delta^n)} \Omega^\bullet_{poly}(\Delta^n)

where moreover the generators dt id t_i are identified with the de Rham differential of the iith barycentric coordinate functions.

This defines a canonical inclusion

Ω poly (Δ n)Ω (Δ smth n) \Omega^\bullet_{poly}(\Delta^n) \hookrightarrow \Omega^\bullet(\Delta^n_{smth})

and there is uniquely the structure of a differential graded-commutative algebra on Ω poly (Δ n)\Omega^\bullet_{poly}(\Delta^n) that makes this a homomorphism of dg-algebras. This is the dg-algebra of polynomial differential forms.

For f:[n 1][n 1]f \colon [n_1] \to [n_1] a morphism of finite non-empty linear orders, let

Ω poly (f):Ω poly (Δ n 2)Ω poly (Δ n 1) \Omega^\bullet_{poly}(f) \;\colon\; \Omega^\bullet_{poly}(\Delta^{n_2}) \to \Omega^\bullet_{poly}(\Delta^{n_1})

be the morphism of dg-algebras given on generators by

Ω poly (f):t i f(j)=it j. \Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j \,.

This yields a simplicial differential graded-commutative algebra

Ω poly (Δ ()):Δ opcdgAlg k \Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k

which is a sub-simplicial object of that of smooth differential form

Ω poly (Δ ())Ω (Δ smth ()). \Omega^\bullet_{poly}(\Delta^{(-)}) \hookrightarrow \Omega^\bullet(\Delta_{smth}^{(-)}) \,.

Piecewise polynomial differential forms

By left Kan extension the functor of polynomial differential forms from def. 3 yields a functor on all simplicial sets

Ω poly :sSetcdgAlg k op. \Omega^\bullet_{poly} \colon sSet \longrightarrow cdgAlg_k^{op} \,.

This is the left adjoint in a nerve and realization adjunction

(Ω poly 𝒦 poly):(dgcAlg ,0) opK polyΩ poly sSet. (\Omega^\bullet_{poly} \dashv \mathcal{K}_{poly}) \;\colon\; (dgcAlg_{\mathbb{Q}, \geq 0})^{op} \underoverset {\underset{K_{poly}}{\longrightarrow}} {\overset{\Omega^\bullet_{poly}}{\longleftarrow}} {\bot} sSet \,.

Composing with the singular simplicial complex functor

Sing:TopsSet Sing \;\colon\; Top \longrightarrow sSet

on topological spaces, this yields a functor on topological spaces

Ω pwpoly :TopSingsSetΩ poly (dgcAlg ,0) op \Omega^\bullet_{pwpoly} \;\colon\; Top \overset{Sing}{\longrightarrow} sSet \overset{\Omega^\bullet_{poly}}{\longrightarrow} (dgcAlg_{\mathbb{Q}, \geq 0})^{op}

which we may think of as assigning “piecewise polynomial” differential forms.

This is the starting point of the Sullivan approach to rational homotopy theory. See there for more

Properties

Let kk be a field of characteristic 0. Let Ω poly :sSetcdgAlg k op\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op} be the left Kan extension of Ω poly :ΔcdgAlg k op\Omega^\bullet_{poly} : \Delta \to cdgAlg_k^{op} from above.

Definition

For SsSetS \in sSet, define a morphism of graded kk-vector spaces

:Ω poly (S)C (S,k) \int : \Omega^\bullet_{poly}(S) \to C^\bullet(S, k)

from polynomial differential forms on simplices to cochains on simplicial sets by sending ωΩ poly n(K)\omega \in \Omega^n_{poly}(K) to the cochain that sends σK n\sigma \in K_n to

σf:= Δ nf max(σ)dt 1dt n, \int_\sigma f := \int_{\Delta^n} f_{max}(\sigma) d t_1 \wedge \cdots d t_n \,,

where on the right we have the ordinary integral of the 1,,n1,\cdots,n-component of the restriction of ff to σ\sigma.

Proposition

The morphism \int is a quasi-isomorphism of cochain complexes.

This is (Bousfield-Gugenheim, theorem 2.2, corollary 3.4).

The following is the central fact of the Sullivan approach to rational homotopy theory:

Proposition

The functor Ω poly \Omega^\bullet_{poly} is the left adjoint of a Quillen adjunction

(Ω poly R):sSetΩ poly RcdgAlg k op (\Omega^\bullet_{poly} \dashv R) \;\colon\; sSet \underoverset {\underset{\Omega^\bullet_{poly}}{\to}} {\overset{R}{\leftarrow}} {\bot} cdgAlg_k^{op}

for the standard model structure on simplicial sets and the projective model structure on commutative dg-algebras.

This is shown in (Bousfield-Gugenheim, section 8).

So in particular Ω poly \Omega^\bullet_{poly} sends cofibrations of simplicial sets to fibrations of dg-algebras. Hence for i:Δ[k]Δ[k]i : \partial \Delta[k] \hookrightarrow \Delta[k] a boundary inclusion the corresponding restriction

i *:Ω poly (Δ k)Ω poly (Δ k) i^* : \Omega^\bullet_{poly}(\Delta^k) \to \Omega^\bullet_{poly}(\partial \Delta^k)

is degreewise surjective.

Proposition

The functor Ω poly \Omega^\bullet_{poly} is a lax monoidal functor whose lax monoidal structure map

X,Y:Ω poly (X)Ω poly (Y)Ω poly (X×Y) \nabla_{X,Y} : \Omega^\bullet_{poly}(X) \otimes \Omega^\bullet_{poly}(Y) \to \Omega^\bullet_{poly}(X \times Y)

is a quasi-isomorphism.

This is reviewed for instance in (Hess, page 12).

Applications

Applications include

References

An original reference is

  • Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2_On PL De Rham Theory and Rational Homotopy Type_ , Memoirs of the A. M. S., vol. 179, 1976.

A standard textbook is

  • Stephen Halperin, Lecture Notes on Minimal Models, Publications de l’U.E.R. Mathématiques Pures et Appliquées, Université des Sciences et techniques, Lille, Vol 3 (1981) Fasc.3.

This is based on

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

A useful survey is in

Revised on February 22, 2017 11:00:13 by Urs Schreiber (147.231.89.7)