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Diffeology Fiber Bundles DIFFEOLOGY FIBER BUNDLES PATRICK IGLESIAS-ZEMMOUR ref. http://math.huji.ac.il/~piz/documents/ShD-lect-DFB.pdf We present in this lecture the theory of diffeology fiber bundles which deviates from the usual locally trivial bundles. Local trivi- ality is replaced in diffeology by local triviality along the plots. A few examples are given and some new situation that happen only in diffeology. The question of what is a fiber bundle in diffeology arises immedi- ately with the case of irrational torus Ta. The direct computation 2 of the first homotopy group shows that the projection T ® Ta behaves like a fibration, with fiber R, but without being locally trivial, since Ta inherits the coarse topology. Indeed, from the diffeology we found directly that p0(Ta) = {Ta} and T˜ a = R with p1(Ta) = Z + aZ ⊂ R, where T˜ a plays the role of universal covering of Ta. So, it was necessary to revise the notion of fiber bundle from clas- sical differential geometry, and adapt it to diffeology in order to include these news objects, specific to diffeology, but without los- ing the main properties of this theory. This is what I have done in my doctoral dissertation in 1985 [Igl85]. The main properties we wanted to preserve was: (1) The homotopy long sequence, that we shall see in the lec- ture on homotopy. Date: December 16, 2020. Shantou 2020 Lecture Notes. 1 2 PATRICK IGLESIAS-ZEMMOUR (2) Any quotient class: G ® G/H is a diffeological fiber bundle, with fiber H, where G is a diffeological group and H any subgroup. These two properties, if satisfied, will explain the direct calculation of the homotopy of the irrational torus. In this lecture we shall see two equivalent definitions of diffeological fiber bundles. The first is pedestrian and operative, involving local triviality along the plots. the second one involving groupoids is more in the spirit of diffeology. Diffeological Fiber Bundles, the Pedestrian Approach 1. Vocabulary [Category of Projections] We call projection any smooth surjection p:Y ® X, with X and Y two diffeological spaces. The space Y is called the total space of the projection and X is called the base. We define a category {Projections} whose obects are the projections and: Definition The morphisms from p:Y ® X to p0 :Y0 ® X0, are the pair of smooth maps (F, f) such that: F Î C (Y, Y0) and f Î C (X, X0), such that 1 1 Y F Y0 0 p p0 with p ◦ F = f ◦ p. 0 X f X The preimage p–1(x) = {y Î Y | p(y) = x} is called the fiber of the projection over x. I use also the word bundle as synonym of projection, which will be specified in some cases in fiber bundle later, when some other properties will be satisfied. Definition An isomorphism from p to p0 will be a pair (F, f) of diffeomorphisms. 2. Definition [Pullbacks of Bundles] Let p:Y ® X be a projection. Let f :X0 ® X be a smooth map. We call the total space of the DIFFEOLOGY FIBER BUNDLES 3 pullback of p by f , or simply the pullback, the space denoted and defined by f ∗(Y) = {(x0, y) Î X0 ´ Y | f (x) = p(y)}. This set is equipped with the subset diffeology of the product X0 ´ Y. pr f ∗(Y) 2 Y pr1 p X0 X f The pullback of p by f is the first projection: ∗ 0 0 0 pr1 : f (Y) ® X with pr1(x , y) = x . 3. Vocabulary [Trivial Projections] Projections on factors are par- ticular cases of projections. We will see often the first projection of direct products: pr1 :X ´ F ® X Definition We say that a projection p:Y ® X is trivial with fiber F if it is isomorphic to the first projection pr1 :X ´ F ® X. It is equivalent to say that there exists an isomorphism with the first projection pr1 :X ´ F ® X with type (F, 1X). X ´ F F Y pr1 p X And that is the way we will use it in general. JProof. Assume that (F, f) is an isomorphism from pr1 :X´F ® X –1 –1 to p:Y ® X. Consider then the isomorphism (f ´ 1F, f ) from pr1 :X ´ F ® X to istelf, we get: f–1 ´ 1 X ´ F F X ´ F F Y pr1 pr1 p X X X f–1 f which gives 4 PATRICK IGLESIAS-ZEMMOUR f–1 ´ 1 (x, y) F (f–1(x), y) F F(f–1(x), y) pr pr1 1 p x f–1(x) f(f–1(x)) = x f–1 f And that is the triangle diagram above. I 4. Vocabulary [Locally Trivial Projections] With trivial projection comes locally trivial projection. Let p:Y ® X a smooth projection as it is defined above. Definition We say that the projection p is locally trivial if there exist a diffeological space F, a D-open covering (Ui)iÎI of X such that the restrictions –1 pi :Yi ® Ui with Yi = p (Ui) and pi = p Yi, are trivial with fiber F. It is equivalent to say that: (1) p is locally trivial at the point x Î X, if there exist a D-open neighborhood U of x such that: the restriction pU :YU ® U –1 is trivial with, with YU = p (U) and pU = p YU. (2) p is locally trivial everywhere with fiber F. 5. Definition [Diffeological Fiber Bundles] A smooth projection p:Y ® X, is a diffeological fibration, or a diffeological fiber budle, if it is locally trivial along the plots, that is, if the pullback of p by any plot P of X is locally trivial with some given fiber F. The space F is the fiber of the fibration. That means precisely the following: For all plots P: U ® X, for all r Î U, there exist an open neigh- ∗ borhood V of r in U such that pr1 : (P V) (Y) ® V is trivial with fiber F, that is, isomorphic to pr1 :V ´ F ® V. Recall that P∗(Y) = {(r, y) Î U ´ Y | P(r) = p(y)}. Note 1. A diffeological fibration p:Y ® X is, in particular, a subduction and even a local subduction. That is, for every plot –1 P:U ® X, for all r Î U and for all y Î Yx = p (x), with x = P(r), there exists a plot Q of Y defined on some open neighborhood V of r lifting P V, that is, P V = p ◦ Q, and such that Q(r) = y. Note 2. There is a hierarchy in the various notions of fiber bundles: DIFFEOLOGY FIBER BUNDLES 5 (1) Trivial fiber bundles are locally trivial (with respect to the D-topology). (2) Locally trivial fiber bundles are locally trivial along the plots. The converse is not true. To be locally trivial along the plots does not mean that the fibra- tion itself is locally trivial, as many examples will point it out. For 2 2 example, the projection of T on the irrational torus Ta = T /Sa is locally trivial along the plots, with fiber R, but not trivial. This is a particular case of quotient G/H of diffeological groups by a subgroup. Note 3. If the base of a diffeological fiber bundle is a manifold, then the fiber bundle is locally trivial. This comes immediately from the definition, consider the pullback by local charts. If moreover the fiber is a manifold, then the diffeological fiber bundle is a fiber bundle in the category of manifolds. This shows in particular that the classical notion of fiber bundle can also be defined directly in diffeological terms as a property of its associated groupoid, as we shall see later, but of course this definition leads to leave an instant the category of manifolds. 6. Proposition [Quotient of Groups by Subgroups] Let G be a diffeological group and H ⊂ G be a subgroup. Then, the projection class: G ® G/H is a diffeological fibration, where H acts on G by left multiplication denoted by L(h)(g) = hg. Note 1. We recall that a diffeological group is a group equipped with a diffeology such that the multiplication (g, g0) 7® gg0 and the inversion g 7® g–1 are smooth. 2 2 Note 2. In particular the projection class: T ® Ta = T /Sa is a diffeological fibration. JProof. Let P: U ® G/H be a plot. We have: P∗(G) = {(r, g) Î U ´ G | class(g) = P(r)}. Let r0 Î U, there exist an open neighborhood V of r0 and a smooth lifting of P, Q: r 7® gr, over V such that class(gr) = P(r). Let ∗ y:V ´ H ® (P V) (G) defined by y(r, h) = (r, hgr). The inverse is given by –1 –1 y :(r, g) 7® (r, ggr ). 6 PATRICK IGLESIAS-ZEMMOUR Since the multiplication and the inversion are smooth, y and y–1 ∗ are smooth and y is an isomorphism from pr1 :V´H ® (P V) (G). Therefore class: G ® G/H is a diffeological fibration. I 7. Proposition [Principal Fiber Bundles] Let X be a diffeological space and g 7® gX be a smooth action of a diffeological group G on X, that is, a smooth homomorphism from G to Diff(X) equipped with the functional diffeology of diffeological group. Let E be the action map, E :X ´ G ® X ´ X with E(x, g) = (x, gX(x)). Proposition. If E is an induction, then the projection class from X to its quotient X/G is a diffeological fibration, with the group G as fiber.
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