nLab differential forms on simplices

Idea

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

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 $n$-simplex)

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

$$\Delta^n_{smth} \;\coloneqq\; \Big\{ (x_0, x_1, \cdots, x_n) \in \mathbb{R}^{n+1} \;\vert\; 0 \leq x_i \leq 1 \;\text{and}\; \underoverset{i = 0}{n}{\sum} x_i \; = 1 \Big\} \hookrightarrow \mathbb{R}^{n+1} \,.$$

For $0 \leq i \leq n$ the function

$$x_i \;\colon\; \Delta^n_{smth} \to \mathbb{R}$$

which picks the $i$th component in the above definition is called the $i$th barycentric coordinate function.

For

$$f \;\colon\; [n_1] \longrightarrow [n_2]$$

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

$$\Delta_{smth}(f) \;\colon\; \Delta^{n_1}_{smth} \longrightarrow \Delta^{n_2}_{smth}$$

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

Definition

(smooth differential forms on the smooth $n$-simplex)

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

Let

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

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

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

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

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

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

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

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

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

$$(\Delta_{smth}(f))^\ast \;\colon\; \Omega^\bullet(\Delta^{n_2}_{smth}) \longrightarrow \Omega^\bullet(\Delta^{n_1}_{smth})$$

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

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

The standard proof of the Poincaré lemma applies to show that

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

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

$$\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},$$

where $b_j$ is as above the $j^{th}$ barycentric coordinate function and each $\Phi_{i_1\ldots i_p}$ is a $C^\infty$-function on $\mathbf{\Delta}^n$.

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

$$(d b_{i_1}\wedge\ldots d b_{i_p})\wedge (d b_{j_1}\wedge\ldots d b_{j_q}) = (d b_{i_1}\wedge\ldots d b_{i_p}\wedge d b_{j_1}\wedge\ldots d b_{j_q})$$

on the generating forms. Now if $f$ is a differentiable function

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

so if

$$\Phi =\sum_{1\leq i_1\lt \ldots\lt i_p\leq n}\Phi_{i_1\ldots i_p}d b_{i_1}\wedge\ldots d b_{i_p},$$

then

$$d\Phi =\sum_{1\leq i_1\lt\ldots\lt i_p\leq n} d\Phi_{i_1\ldots i_p} \wedge d b_{i_1} \wedge \ldots d b_{i_p},$$

Polynomial differential forms

Definition

For $n \in \mathbb{N}$ write

$$\Omega_{poly}^{\bullet}(\Delta^n) \;\coloneqq\; Sym^\bullet_{\mathbb{Q}} \langle t_0, \cdots, t_n, d t_0, \cdots, d t_n\rangle/\left(\sum t_i -1, \sum d t_i \right)$$

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

In particular, in degree 0 these are called the polynomial functions

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

due to the canonical inclusion

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

into the smooth functions on the $n$-simplex according to def. , obtained by regarding the generator $t_i$ as the $i$th barycentric coordinate function.

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

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

where moreover the generators $d t_i$ are identified with the de Rham differential of the $i$th barycentric coordinate functions.

This defines a canonical inclusion

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

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

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

$$\Omega^\bullet_{poly}(f) \;\colon\; \Omega^\bullet_{poly}(\Delta^{n_2}) \to \Omega^\bullet_{poly}(\Delta^{n_1})$$

be the morphism of dg-algebras given on generators by

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

This yields a simplicial differential graded-commutative algebra

$$\Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k$$

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

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

Piecewise polynomial differential forms

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

$$\Omega^\bullet_{poly} \colon sSet \longrightarrow cdgAlg_k^{op} \,.$$

This is the left adjoint in a nerve and realization adjunction

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

Composing with the singular simplicial complex functor

$$Sing \;\colon\; Top \longrightarrow sSet$$

on topological spaces, this yields a functor on topological spaces

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

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

Definition

For $S \in sSet$, define a morphism of graded $k$-vector spaces

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

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

$$\int_\sigma f := \int_{\Delta^n} f_{max}(\sigma) d t_1 \wedge \cdots d t_n \,,$$

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

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 $\Omega^\bullet_{poly}$ is the left adjoint of a Quillen adjunction

$$(\Omega^\bullet_{poly} \dashv R) \;\colon\; sSet \underoverset {\underset{\Omega^\bullet_{poly}}{\to}} {\overset{R}{\leftarrow}} {\bot} cdgAlg_k^{op}$$

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

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

is degreewise surjective.

Proposition

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

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

is a quasi-isomorphism.

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

Applications

Applications include

• the Sullivan construction in rational homotopy theory;

• the sSet-enrichment of the model structure on dg-algebras over an operad.

• higher Lie integration

Related concepts

PL de Rham complex

References

Original articles:

• Dennis Sullivan, p. 297 (30 of 64) of: Infinitesimal computations in topology, Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam:PMIHES_1977__47__269_0)

• Aldridge Bousfield, Victor Gugenheim, p. 1 of: On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)

• Phillip Griffiths, John Morgan, p. 83-84 of: Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (2013) (doi:10.1007/978-1-4614-8468-4)

Review:

• Yves Félix, Stephen Halperin, Jean-Claude Thomas, p. 122 (154 of 574) in: Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (doi:10.1007/978-1-4613-0105-9)

• Kathryn Hess, Def. 1.19 (p. 8) of: Rational homotopy theory: a brief introduction, contribution to Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago (arXiv:math.AT/0604626), chapter in Luchezar Lavramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007 (doi:10.1090/conm/436)