and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
de Rham space, crystal?
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Write
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).
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:
(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:
Write
for the simplicial object in dgc-algebras (over the real numbers) given by smooth differential forms on simplices.
(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:
This receives an evident inclusion from the PL de Rham complex (4):
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
which restricts to a diffeomorphism onto its image in the interior of any simplex
there is a weak homotopy equivalence of simplicial sets
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).
(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 $X$:
Here $S(X)$ is the simplicial complex corresponding to any smooth triangulation of $X$.
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.
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)
Last revised on July 7, 2021 at 12:15:15. See the history of this page for a list of all contributions to it.