Notes on Diffeological Bundles

Enxin Wu

Abstract

This is an extended notes of a presentation for the Stacks Seminar organized by Jeffrey Morton in the University of Western Ontario on March 19, 2010.

Contents

1 Diffeological spaces 1 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Diffeological spaces and some basic properties . . . . . . . . . . . . . . . . . 2

2 Diffeological bundles 3 2.1 Diffeological bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 An example – the irrational torus . . . . . . . . . . . . . . . . . . . . . . . . 5

1 Diffeological spaces

1.1 Motivations

Manifolds are very popular objects in mathematics, on which many different branches of mathematics meet – analysis, algebra, geometry, topology, etc.

However, the category of manifolds are not that pleasant – not every subset or quotient set of a manifold is a manifold; many nice infinite dimensional spaces, including the function spaces, are hard to have canonical classical manifold structures, etc.

Now a natural question would be how to find a “nice” (bicomplete, have internal hom functor, etc) category, which contains the category of manifolds as a full subcategory, on which you could still do differential geometry, differential topology, etc.

By checking the definition of a manifold carefully, you can see the key part is to have a chart – which is a homeomorphism from an open subset of the manifold to an open subset of a Euclidean space. In fact, the set of all “mapping-in” and the set of all “mapping-out” determines each other. For more detailed explanations of these, check [S].

There are several ways to “generalize” a manifold, basically there are three types – use both “mapping-in” and “mapping-out” together with a compatibility, or use “mapping-in” only, or use “mapping-out” only. For the comparison of many existing smooth spaces, we refer the reader to [S].

In this notes, we focus on a kind of “mapping-in” smooth space called diffeological spaces, which was initiated by Jean-Marie Souriau in the 1980’s.

With this tool, we might be able to deal with some interesting pathological spaces, for example the irrational torus, in a non-traditional way.

1.2 Diffeological spaces and some basic properties

Definition 1.1. A diffeological space (X, D) is a set X together with a set of maps D = {U → X | U open in ℝ^{n} for some n ∈ ℕ}, satisfying the following three axioms:

(1) (Covering) Every constant map belongs to D;

(2) (Sheaf Condition) If ϕ : U → X with U open in some ℝ^{n} is locally in D, then ϕ ∈ D;

(3) (Smooth Compatibility) If U → X belongs to D, and V → U is smooth with V open in some ℝ^{m}, then the composition is also in D.

Note the definition of a diffeological space does Not require it to be a topological space to start with.

Definition 1.2. Let (X, D_{X}) and (Y, D_{Y}) be two diffeological spaces, and let f : X → Y be a set map. We call f a smooth map, if f(D_{X}) ⊆ D_{Y}.

An element in D_{X} is called a plot of the diffeological space (X, D_{X}). Without ambiguity, we will write a diffeological space X instead of (X, D_{X}).

Diffeological spaces together with smooth maps form a category.

Here is a list of basic properties of this category:

1. Fix a set X, the set of all diffeologies on X forms a complete lattice. Hence we can talk about diffeology generated by some plots, and define dimension of a diffeological space.

2. This category contains the category of manifold as a full subcategory.

3. This category is bicomplete, and we can define arbitrary sub/quotient diffeological spaces. The limit/subspace (colimit/quotient space) can be given as a limit/subset (colimit/quotient set) of the underlying set, together with the largest (smallest) diffeology making all required maps smooth.

4. There is a natural diffeology for the function space, with which, the category is (locally) cartesian closed.

5. [BH] This category is equivalent to the category of concrete sheaves over a certain concrete site.

6. [CW, Is] This category is locally presentable.

7. [CW] There is an adjoint pair from this category to the category of topological spaces, with the image of the left adjoint functor exactly those so called Δ-generated spaces.

8. [La, I2] Tangent space, differential forms, de Rham cohomology, etc, can be defined on any diffeological spaces.

9. [I1, I2] Smooth (relative) homotopy groups π_{n}^{D} can be defined recursively on any pointed diffeological space, by loop spaces, which is a pointed set for i = 0(1), a group for i = 1(2), and an abelian group for i = 2(3). There is a long exact smooth (relative) homotopy groups sequence for a pair of pointed spaces.

For detailed explanations of these properties, we refer readers to the corresponding references.

Remark 1.3. If X is a manifold, then the diffeology generated by all charts is called the standard diffeology on X. Then dimension, D-topology (the largest topology on X making all plots continuous), tangent space, differential forms, de Rham cohomology, smooth homotopy groups, etc, are the same as those defined in the usual sense.

2 Diffeological bundles

2.1 Diffeological bundles

Definition 2.1. A diffeological groupoid K is a groupoid object in the category of diffeological spaces, that is, a groupoid K such that Obj(K) and Mor(K) are two diffeological spaces, and i : Obj(K) → Mor(K) given by i(a) = id_{a}, s, t : Mor(K) → Obj(K) given by s(f : a → b) = a and t(f : a → b) = b, j : Mor(K) → Mor(K) given by j(f : a → b) = (f^{−1} : b → a), and c : A → Mor(K) given by c(g, f) = g ∘ f, where A is the pullback of the diagram

\xymatrix{ & Mor(K) \ar[d]^{t} \\ Mor(K) \ar[r]_{s} & Obj(K),}

equipped with the sub-diffeology, are all smooth maps.

Definition 2.2. A diffeological groupoid K is said to be perfect, if (s, t) : Mor(K) → Obj(K) × Obj(K) is a subduction, that is, it is surjective and the product diffeology on Obj(K) × Obj(K) is exactly the same as the smallest diffeology on Obj(K) × Obj(K) making the given map smooth.

Proposition 2.3. Given a smooth map f : A → Y, we can construct an associated diffeological groupoid K_{f} with Obj(K_{f}) = Y, Mor_{K_{f}} (y, y′) = {diffeomorphisms f^{−1}(y) → f^{−1}(y′) with each of them equipped with the sub-diffeology from X} together with the smallest diffeology making all required maps and the evaluation map B → X smooth, where B is the pullback of the following diagram:

\xymatrix{ & X \ar[d]^{f} \\ Mor(K_{f}) \ar[r]_{s} & Y = Obj(K_{f}).}

Proof. Not hard to see that it is enough to equip Mor(K_{f}) with the smallest diffeology from i : Obj(K_{f}) → Mor(K_{f}) for all required maps except the evaluation map to be smooth, that is, U → Mor(K_{f}) is a plot if and only if locally it is constant or locally it factors through i. Then we can give B the smallest diffeology D_{B} generated by smallest diffeologies on B making the evaluation map and the two natural maps from the pullback diagram smooth respectively, where Mor(K_{f}) is equipped with the diffeology described above. Finally give Mor(K_{f}) the smallest diffeology making B → Mor(K_{f}) in the pullback diagram smooth, where B is equipped with diffeology D_{B}. The rest are easy.

Definition 2.4. A diffeological bundle is a smooth surjective map f : X → Y, such that the associated diffeological groupoid K_{f} is perfect. In this case, we call X the total space, Y the base space, and F = f^{−1}(x) equipped with the sub-diffeology of X the fibre at x ∈ X.

Remark 2.5. The perfectness condition says that, for any diffeological bundle, fibres at any two points are diffeomorphic.

Definition 2.6. A diffeological bundle f : X → Y is called trivial, if there exists some diffeological space F, and a diffeomorphism h : X → F × Y, where F × Y is equipped with the product diffeology, such that the following diagram is commutative:

\xymatrix{X \ar[dr]_{f} \ar[r]^{h} & F × Y \ar[d]^{pr_{2}} \\  & Y.}

Theorem 2.7. Let f : X → Y be a smooth surjective map between two diffeological spaces. The following three statements are equivalent:

1. f : X → Y is a diffeological bundle;

2. There exists a diffeological space F such that the pullback of f along any plot of Y is locally trivial of fibre type F;

3. There exists a diffeological space F such that the pullback of f along any global plot of Y (that is, plot of the form ℝ^{n} → Y for some n ∈ ℕ) is trivial of fibre type F.

Proof. 1 ⇒ 2:

Fix y ∈ Y, and write f^{−1}(y) = F. Let ϕ : U → Y be a plot of Y. Then y × ϕ : U → Y × Y = Obj(K_{f}) × Obj(K_{f}) is a plot of Y × Y, hence locally this map factors through Mor(K_{f}) → Y × Y. Without loss of generality, we still use the same U and ϕ for the local data, and write the lifting by ψ : U → Mor(K_{f}). Write C for the pullback of the diagram

\xymatrix{ & X \ar[d]_{f} \\ U \ar[r]_{ϕ} & Y.}

We have a smooth map U × F → C given by (u, x) ↦ (u, ϕ(u)(x)), which has a smooth inverse C → U × F given by (u, x) ↦ (u, (ϕ(u))^{−1}(x)).

2 ⇒ 3:

This is essentially the next proposition.

3 ⇒ 1:

Let (ϕ, ψ) : U → Y × Y be a plot of Y × Y, then ϕ : U → Y and ψ : U → Y are both plots of Y. Hence, locally, the pullbacks of f : X → Y along ϕ|_{U′} : U′ → Y and ψ|_{U′′} : U′′ → Y are U′ × F and U′′ × F, respectively. Then we can define the lifting of (ϕ, ψ) along Mor(K_{f}) → Y × Y by using the transition function, and check it is smooth.

Proposition 2.8. Any diffeological bundle with base space ℝ^{n} is trivial.

Proof. See [I1].

Theorem 2.9. Let f : X → Y be a diffeological bundle of fibre type F = f^{−1}(y) (equipped with sub-diffeology of X) for some y ∈ Y. Pick x ∈ F. Then f induces an isomorphism π_{i}^{s}(X, F, x) → π_{i}^{s}(Y, y) of pointed sets for i = 1; of groups for i ≥ 2. Therefore, we have the following long exact sequence:

\xymatrix{⋯ \ar[r] & π_{n}^{D}(F, x) \ar[r]^{i_{∗}} & π_{n}^{D}(X, x) \ar[r] & π_{n}^{D}(Y, y) \ar[r] & π_{n−1}^{D}(F, x) \ar[r] & ⋯ \ar[r] & π_{0}^{D}(M) \ar[r] & 0.}

Proof. See [I1].

2.2 An example – the irrational torus

Definition 2.10. A diffeological group G is a group object in the category of diffeological spaces, that is, it is both a diffeological space and a group, such that the multiplication and inverse maps are smooth. A sub-diffeological group H of a diffeological group G is both a subset of G and a diffeological group, whose group and diffeological structure is inherited from those of G.

Theorem 2.11. Let G be a diffeological group, and let H be a sub-diffeological group. Then G → G/H is a diffeological bundle of fibre type H.

Proof. Assume G/H is left coset {gH, g ∈ G} with quotient diffeology from π : G → G/H. Let ϕ : U → G/H be a plot, then locally it factors through π : G → G/H, that is, for any u ∈ U, there exists a neighborhood i : V ↪ U and a smooth map ψ : V → U such that π ∘ ψ = ϕ ∘ i. As a set, the pullback R of the diagram

\xymatrix{ & G \ar[d]^{π} \\ V \ar[r]_{ϕ∘i} & G/H}

is V ×H, so we need to prove that the diffeology matches. Define (pr_{1}, β) : V ×H → V ×G by (v, h) ↦ (v, ψ(v)h). It is smooth since multiplication is smooth, and π ∘ β = ϕ ∘ i ∘ pr_{1}. Hence we have a smooth map V ×H → R by (pr_{1}, β). Now we define its inverse R → V ×H by (v, h) ↦ (v, ψ(v)^{−1}h). It is smooth since both multiplication and inverse are smooth.

Example 2.12. Let T^{2} = ℝ^{2}/ℤ^{2} be the torus with the standard diffeology, and ℝ_{θ} be an

irrational line in the torus (that is, the image of \xymatrix{ℝ \ar[r]^{f} & ℝ^{2} \ar[r]^{π} & ℝ^{2}/ℤ^{2}} in T^{2}, where f(x) = (x, θx) with θ an irrational number) with sub-diffeology. Then both of T^{2} and ℝ_{θ} are diffeological groups under addition, and ℝ_{θ} is in fact a sub-diffeological group of T^{2}. Hence T^{2} → T^{2}/ℝ_{θ} is a diffeological bundle of fibre type ℝ_{θ} by the above theorem. Not hard to see that ℝ_{θ} is isomorphic to ℝ with the standard diffeology, and for any manifold viewed as a diffeological space with the standard diffeology, the smooth homotopy groups are isomorphic to the usual homotopy groups. From the long exact sequence, we see that π_{1}^{D}(T^{2}/ℝ_{θ}) = ℤ ⊕ ℤ, while in the usual topology π_{1}(T^{2}/ℝ_{θ}) = 0 since ℝ_{θ} is dense in T^{2}.

References

[BH] John C. Baez and Alexander E. Hoffnung, Convenient Categories of Smooth Spaces, available at http://math.ucr.edu/home/baez/smooth.pdf.

[CW] Dan Christensen and Enxin Wu, Notes on Diffeological Spaces, in preparation.

[I1] Patrick Iglesias-Zemmour, Fibrations diffeologiques et Homotopie, These de Doctorat Es-sciences, L’Universite de Prouence, 1985.

[I2] Patrick  Iglesias-Zemmour,  Diffeology,  draft  available  at

http://math.huji.ac.il/∼piz/documents/Diffeology.pdf.

[Is] Samuel B. Isaacson, Concrete Sheaves are Locally Presentable, email contact.

[La] Martin Laubinger, Diffeological Spaces, Proyecciones Journal of Mathematics, Vol. 25, No. 2, 2006, pp. 151-178.

[S]  Andrew  Stacey,  Comparative  Smootheology,  available  at

www.math.ntnu.no/∼stacey/documents/smthcat.ps.