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
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
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 sSet a simplicial set, its PL de Rham complex is the hom-object of simplicial objects from to (1), hence is the following end in dgcAlgebras, here over the real numbers:
(Bousfield-Gugenheim 76, Sec. 2, p. 7)
For 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 sSet a simplicial set, its PS de Rham complex (“piecewise smooth”) is the hom-object of simplicial objects from to (3), hence is the following end in dgcAlgebras:
This receives an evident inclusion from the PL de Rham complex (4):
For a smooth manifold, and 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 (this being the adjunction unit of the 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 preserves all weak equivalences, and those between CW-complexes, by Ken Brown's lemma).
(PL de Rham complex of smooth manifold is equivalent to de Rham complex)
Let 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) and the smooth de Rham complex of :
Here is the simplicial complex corresponding to any smooth triangulation of .
For the two morphisms on the right this is Griffith-Morgan 13, Cor. 9.9.
For the morphism on the left this follows since is a weak homotopy equivalence and since , 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)
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