A homotopy theory of diffeological spaces

A homotopy theory of diffeological spaces Dan Christensen and Enxin Wu MIT Jan. 5, 2012

Motivation Smooth manifolds contain a lot of geometric information: tangent spaces, differential forms, de Rham cohomology, etc. These can be put to great use in proving results. But the category of smooth manifolds is not closed under many useful constructions, such as subspaces, quotients and function spaces. The category of (weak Hausdorff compactly generated) topological spaces is closed under these operations, but the geometric information is missing. Can we have the best of both worlds?

Diffeological spaces Definition (J. Souriau, 1980). A diffeological space is a set X together with a specified family of maps U X (called plots) for each open U R n and each n, such that for every open U, V R n : (1) Every constant map U X is a plot; (2) If U X is a plot and V U is smooth, then V U X is a plot; (3) If U = i U i is an open cover and U X is a map such that every restriction U i X is a plot, then U X is a plot. Definition. For diffeological spaces X and Y, a function X Y is smooth if for every plot U X in X, U X Y is a plot in Y. Definition. The category of diffeological spaces with smooth maps is denoted by Diff.

Properties of Diff The category of smooth manifolds is a full subcategory of Diff: Each smooth manifold M is a diffeological space: a map U M is declared to be a plot iff it is smooth in the usual sense. A function M N between smooth manifolds is smooth in the usual sense iff it is smooth as a smooth map of diffeological spaces. Remark. Kriegl-Michor s result says this is also true for smooth manifolds with boundary.

Properties of Diff, II Diff is closed under limits and colimits. For example, if Y is a subset of a diffeological space X, we can declare a function U Y to be a plot if the composition U Y X is a plot. In this case, Y is called a sub-diffeological space of X. If Y is a quotient of a diffeological space X, we can declare a function U Y to be a plot if locally it is of the form U i X Y, with the first map a plot of X. In this case, Y is called a quotient diffeological space of X. If X and Y are diffeological spaces, say that U X Y is a plot if each of U X and U Y is.

Properties of Diff, III Diff is cartesian closed: write Diff(X, Y ) for the set of smooth maps from X to Y. This is a diffeological space with a function U Diff(X, Y ) defined to be a plot if the natural map U X Y is smooth. One can show that is a diffeomorphism. Diff(W X, Y ) Diff(W,Diff(X, Y )) Given a diffeological space, we still have geometric information: dimension, differential forms, de Rham cohomology, tangent space at a point, tangent bundles, etc (Laubinger, Iglesias-Zemmour, etc), which are generalizations for smooth manifolds. Diff is locally presentable, but Top is not.

Examples of diffeological spaces For any diffeological space X, the loop space Diff(S 1, X ) and the path space Diff(I, X ) are diffeological spaces. Such examples are a big motivation for generalizing manifolds. A cross in R 2 with the following diffeologies: (1) the gluing diffeology. Then any smooth curve making a turn must stop for a finite amount of time. We call this a border crossing; (2) the sub-diffeology. Then there exists a curve which makes a turn but only halts for an instant. We call this a stop sign.

Naive homotopy theory of diffeological spaces In his thesis, Iglesias-Zemmour 85 defined the smooth homotopy groups π D n (X, x 0 ) of a pointed diffeological space (X, x 0 ) inductively, using loop spaces. As usual, π D 0 is a set, πd 1 is a group, and πd n is abelian for n 2. For a smooth manifold, the smooth and topological homotopy groups agree by smooth approximation. Every diffeological space has a natural topology (called the D-topology), but we will see that in general the smooth and topological homotopy groups do not agree.

Diffeological bundles [Iglesias-Zemmour 85] Definition. A diffeological bundle is a smooth surjective map X Y such that the pullback along any plot of Y is locally trivial. Theorem. If X Y is a diffeological bundle with fibre F, then there is a long exact sequence πn D (F ) πn D (X ) πn D (Y ) πn 1 D (F ) πd 0 (Y ). Theorem. If H is any subgroup of a diffeological group G, then G G/H is a diffeological bundle with fibre H. Example. Let H be a line of irrational slope in the torus T 2 and let A = T 2 /H. Then T 2 A is a diffeological bundle. From the long exact sequence, π D 1 (A) = Z Z. But π 1(A) = 0. All of this and much more is detailed in Iglesias-Zemmour s book in progress, which is freely available online.

A model category candidate for Diff [Work in progress with my supervisor, Dan Christensen. Many variations are being considered as well.] Let A n = {(x 0,, x n ) R n+1 x i = 1}, with the sub-diffeology. Then A n = R n. These A n s behave formally like the standard simplices n, and they form a cosimplicial object. For formal reasons, we get an adjoint pair : sset Diff: S called diffeological realization and smooth singular functors. We have n = A n, that preserves colimits and that S(X ) n =Diff(A n, X ).

A model category candidate for Diff, II We define a map X Y in Diff to be a weak equivalence (fibration) if S(X ) S(Y ) is a weak equivalence (fibration) in sset. A map X Y is a cofibration if it has the left lifting property with respect to all trivial fibrations. We haven t yet completed the proof that these form a model category, but we have some partial results.

Fibrant objects An important first step is to study the fibrant objects. By definition, these are the diffeological spaces such that S(X ) is a fibrant simplicial set, i.e. a Kan complex. Proposition (Christensen-W.) If X is fibrant, then π D (X ) = π (SX ). Corollary If X and Y are fibrant, then a map X Y is a weak equivalence iff π D (X ) π D (Y ) are isomorphisms for all basepoints. Unfortunately, not all objects are fibrant. For example, the cross in R 2 with the gluing diffeology and the sub-diffeology are both not fibrant.

Theorem (Christensen-W.) Every homogeneous space is fibrant. Corollary (Christensen-W.) Every smooth manifold without boundary is fibrant.

Theorem (Christensen-W.) Every homogeneous space is fibrant. Proof This follows from the following results: Every diffeological group is fibrant (in Diff), since every simplicial group is fibrant (in sset). Lemma (Christensen-W.) A diffeological bundle with fibrant fibre is a fibration. Lemma (Christensen-W.) If X Y is a diffeological bundle with X fibrant, then Y is fibrant. Corollary (Christensen-W.) Every smooth manifold without boundary is fibrant.