diffeology fiber bundlesmath.huji.ac.il/~piz/documents/shd-lect-dfb.pdf · 2020. 12. 16. ·...

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 bundleswhich deviates from the usual locally trivial bundles. Local trivi-ality is replaced in diffeology by local triviality along the plots. Afew examples are given and some new situation that happen onlyin diffeology.

The question of what is a fiber bundle in diffeology arises immedi-ately with the case of irrational torus Tα. The direct computationof the first homotopy group shows that the projection T2 → Tαbehaves like a fibration, with fiber R, but without being locallytrivial, since Tα inherits the coarse topology. Indeed, from thediffeology we found directly that

π0(Tα) = {Tα} and T̃α = R with π1(Tα) = Z + αZ ⊂ R,

where T̃α plays the role of universal covering of Tα.

So, it was necessary to revise the notion of fiber bundle from clas-sical differential geometry, and adapt it to diffeology in order toinclude these news objects, specific to diffeology, but without los-ing the main properties of this theory. This is what I have donein 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 anysubgroup.

These two properties, if satisfied, will explain the direct calculationof the homotopy of the irrational torus.

In this lecture we shall see two equivalent definitions of diffeologicalfiber bundles. The first is pedestrian and operative, involving localtriviality along the plots. the second one involving groupoids ismore in the spirit of diffeology.

Diffeological Fiber Bundles, the Pedestrian Approach

1. Vocabulary [Category of Projections] We call projection anysmooth surjection π : Y → X, with X and Y two diffeological spaces.The space Y is called the total space of the projection and X iscalled the base. We define a category {Projections} whose obectsare the projections and:

Definition The morphisms from π : Y → X to π′ : Y′ → X′, are thepair of smooth maps (Φ, φ) such that:

Φ ∈ C∞(Y,Y′) and φ ∈ C∞(X,X′),such that

Y Y′

X X′

π

Φ

π′

φ

with π′ ◦ Φ = φ ◦ π.

The preimage

π–1(x) = {y ∈ Y | π(y) = x}

is called the fiber of the projection over x. I use also the wordbundle as synonym of projection, which will be specified in somecases in fiber bundle later, when some other properties will besatisfied.

Definition An isomorphism from π to π′ will be a pair (Φ, φ) ofdiffeomorphisms.

2. Definition [Pullbacks of Bundles] Let π : Y → X be a projection.Let f : X′ → X be a smooth map. We call the total space of the

DIFFEOLOGY FIBER BUNDLES 3

pullback of π by f , or simply the pullback, the space denoted anddefined by

f ∗(Y) = {(x ′, y) ∈ X′ × Y | f (x) = π(y)}.

This set is equipped with the subset diffeology of the productX′ × Y.

f ∗(Y) Y

X′ X

pr1

pr2

π

f

The pullback of π by f is the first projection:

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 projectionof direct products:

pr1 : X × F → XDefinition We say that a projection π : Y → X is trivial with fiberF if it is isomorphic to the first projection pr1 : X × F → X.It is equivalent to say that there exists an isomorphism with thefirst projection pr1 : X × F → X with type (Φ, 1X).

X × F Y

Xpr1

Φ

π

And that is the way we will use it in general.

JProof. Assume that (Φ, φ) is an isomorphism from pr1 : X×F → Xto π : Y → X. Consider then the isomorphism (φ–1 × 1F, φ–1) frompr1 : X × F → X to istelf, we get:

X × F X × F Y

X X X

pr1

φ–1 × 1F

pr1

Φ

π

φ–1 φ

which gives


(x, y) (φ–1(x), y) Φ(φ–1(x), y)

x φ–1(x) φ(φ–1(x)) = x

pr1

φ–1 × 1F

pr1

Φ

π

φ–1 φ

And that is the triangle diagram above. I4. Vocabulary [Locally Trivial Projections] With trivial projectioncomes locally trivial projection. Let π : Y → X a smooth projectionas it is defined above.Definition We say that the projection π is locally trivial if thereexist a diffeological space F, a D-open covering (Ui)i∈I of X suchthat the restrictions

πi : Yi → Ui with Yi = π–1(Ui) and πi = π � Yi,

are trivial with fiber F.It is equivalent to say that:

(1) π is locally trivial at the point x ∈ X, if there exist a D-openneighborhood U of x such that: the restriction πU : YU → Uis trivial with, with YU = π–1(U) and πU = π � YU.

(2) π is locally trivial everywhere with fiber F.

5. Definition [Diffeological Fiber Bundles] A smooth projectionπ : 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 πby any plot P of X is locally trivial with some given fiber F. Thespace F is the fiber of the fibration.That means precisely the following:For all plots P: U → X, for all r ∈ U, there exist an open neigh-borhood V of r in U such that pr1 : (P � V)

∗(Y) → V is trivial withfiber F, that is, isomorphic to pr1 : V × F → V. Recall that

P∗(Y) = {(r, y) ∈ U × Y | P(r) = π(y)}.

Note 1. A diffeological fibration π : Y → X is, in particular, asubduction and even a local subduction. That is, for every plotP: U → X, for all r ∈ U and for all y ∈ Yx = π–1(x), with x = P(r),there exists a plot Q of Y defined on some open neighborhood Vof r lifting P � V, that is, P � V = π ◦Q, and such that Q(r) = y.Note 2. There is a hierarchy in the various notions of fiber bundles:


(1) Trivial fiber bundles are locally trivial (with respect to theD-topology).

(2) Locally trivial fiber bundles are locally trivial along theplots. 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. Forexample, the projection of T2 on the irrational torus Tα = T2/Sαis locally trivial along the plots, with fiber R, but not trivial. Thisis a particular case of quotient G/H of diffeological groups by asubgroup.

Note 3. If the base of a diffeological fiber bundle is a manifold, thenthe fiber bundle is locally trivial. This comes immediately fromthe definition, consider the pullback by local charts. If moreoverthe fiber is a manifold, then the diffeological fiber bundle is a fiberbundle in the category of manifolds. This shows in particular thatthe classical notion of fiber bundle can also be defined directly indiffeological terms as a property of its associated groupoid, as weshall see later, but of course this definition leads to leave an instantthe category of manifolds.

6. Proposition [Quotient of Groups by Subgroups] Let G be adiffeological group and H ⊂ G be a subgroup. Then, the projectionclass : G → G/H is a diffeological fibration, where H acts on G byleft multiplication denoted by L(h)(g) = hg.

Note 1. We recall that a diffeological group is a group equippedwith a diffeology such that the multiplication (g, g ′) 7→ gg ′ andthe inversion g 7→ g–1 are smooth.Note 2. In particular the projection class : T2 → Tα = T2/Sα is adiffeological fibration.

JProof. Let P: U → G/H be a plot. We have:

P∗(G) = {(r, g) ∈ U × G | class(g) = P(r)}.

Let r0 ∈ U, there exist an open neighborhood V of r0 and a smoothlifting of P, Q: r 7→ gr , over V such that class(gr) = P(r). Let

ψ : V × H → (P � V)∗(G) defined by ψ(r,h) = (r,hgr).

The inverse is given by

ψ–1 : (r, g) 7→ (r, gg–1r ).


Since the multiplication and the inversion are smooth, ψ and ψ–1

are smooth and ψ is an isomorphism from pr1 : V×H → (P � V)∗(G).Therefore class : G → G/H is a diffeological fibration. I7. Proposition [Principal Fiber Bundles] Let X be a diffeologicalspace and g 7→ gX be a smooth action of a diffeological group G onX, that is, a smooth homomorphism from G to Diff(X) equippedwith 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 Xto its quotient X/G is a diffeological fibration, with the group Gas fiber. We shall say that the action of G on X is principal.

Definition. If a projection π : X → Q is equivalent to class : X →G/H,

X

X/G Q

class π

φ

that is, if there exists a diffeomorphism φ : G/H → Q such thatπ = φ ◦ class, then we shall say that π is a principal fibration, or aprincipal fiber bundle, with structure group G.

Note a). If the action E is inductive, then it is in particular in-jective, which implies that the action of G on X is free, which isindeed a necessary condition.

Note b). The quotients class : G → G/H are principal bundles withgroup H.

Note c). Let π : X → Q the G-principal fiber bundle, then thepullback of π by any plot P: U → Q is locally trivial, let say withΦ : V × G → (P � V)∗(X)

V × G (P � V)∗(X)

Vpr1

Φ

pr1

and writeΦ(r, g) = (r, Φr(g)).


Then, the isomorphism Φ satisfies the equivariant property:

Φr(gg ′) = gX(Φr(g ′)).

In particular,

Φr(g) = gX(φ(r)) with φ(r) = Φr(1G).

And φ is a local lifting of the plot P

φ : V → X and π ◦ φ = P � V.

X

V U Q

πφ

P

Note d). A principal fiber bundle is trivial if and only if it admitsa global smooth section.

Diffeological Fiber Bundles, the Groupoid Approach

8. Definition [Structure Groupoid of a Projection] Let π : Y → Xbe a smooth surjection. Let us define

Obj(K) = X and for all x, x ′ ∈ X, MorK(x, x ′) = Diff(Yx, Yx ′),

where the Yx = π–1(x), x ∈ X, are equipped with the subsetdiffeology. Let us define on

Mor(K) =⋃

x,x ′∈XMorK(x, x

′)

the natural multiplication f · g = g ◦ f , for f ∈ MorK(x, x ′) andg ∈ MorK(x ′, x ′′), K is clearly a groupoid. The source and targetmaps are given by

src(f ) = π(def(f )) and trg(f ) = π(Val(f )).

The groupoid K is then equipped with a functional diffeology of Kdefined as follows. Let Ysrc be the total space of the pullback of πby src, that is,

Ysrc = {(f , x) ∈ Mor(K) × Y | x ∈ def(f )}.

We define the evaluation map ev as usual:

ev : Ysrc → Y with ev(f , x) = f (x).


b = src(f ) b' = trg(f )

Xb = def(f ) val(f ) = Xb'

Diff(Xb ,Xb')

B

X

f

g!∈!Ejgg)Zy-Zy◢*

Zy!>!Efg)g* Zy◢!>!Wbm)g*

y!>!tsd)g* y◢!>!ush)g*Y

╮ ╮

Figure 1. The groupoid associated with a projection.

There exists a coarsest diffeology on Mor(K), which gives to K thestructure of a diffeological groupoid and such that the evaluationmap ev is smooth. It will be called again the functional diffeology.Equipped with this functional diffeology, the groupoid K capturesthe smooth structure of the projection π. It is why we define K asthe structure groupoid of the surjection π.Note 1. If X is reduced to a point, Obj(K) = {?}, this diffeologycoincides with the usual functional diffeology of Diff(Y) = Mor(K).Note 2. This construction also applies when we have just a par-tition P on a diffeological space Y. We can equip the quotientQ = Y/P with the quotient diffeology, and we get the structuregroupoid of the partition as the structure groupoid of the projec-tion π : Y → Q.9. Definition [Fibrating Groupoid] Let π : Y → X be a smoothprojection.The projection π is a diffeological fibration if the structure groupoidK is fibrating, that is, if and only if the characteristic map

χ : Mor(K) → B × Bis a subduction. In particular, the preimages Yx = π–1(x), arenecessarily all diffeomorphic since χ is surjective.


This definition is completely equivalent to the previous one §5.

Associated Fiber Bundles

10. Definition [Associated Fiber Bundles] Consider a principalfiber bundle π : Y → X with group G. Let E be a diffeological spacewith a smooth action of G, that is a smooth morphism g → gEfrom G to Diff(E). Let the product over X be the quotient space

Y ×G E = (Y × E)/G,

where G acts on the product diagonally

gY×E(y, e) = (gY(y), gE(e)).

Then, the projection:

pr(class(y, e)

)= π(y),

from Y ×G E to X is a diffeological fiber bundle with fiber E.

Y × E Y ×G E

Y X

pr1

class

pr

π

11. Proposition [Main Theorem] Let π : Y → X be a diffeologicalfiber bundle with fiber E, then there exists a principal fiber bundlepr: T → X with some diffeological group G acting smoothly onE, such that π is associate to pr. Actually, one can always takeG = Diff(E).

Covering Diffeological Spaces

A special case of fiber bundle plays a special role in differentialgeometry, the covering spaces.

12. Definition [Covering spaces] Let X be a connected space, thatis, for all pair of points x, x ′ ∈ X, there exists a smooth pathγ ∈ Paths(X) = C∞(R, X) such that γ(0) = x and γ(1) = x ′. Wesay that x and x ′ are connected.

We call covering over X any fiber bundle such that the fiber isdiscrete (of course diffeologically). The covering may be non con-nected but we focus in general on connected coverings.


The main theorem about covering is about the universal covering..13. Proposition [Universal Covering] Let X be a connected dif-feological space, there exists a unique, up to equivalence, simplyconnected which is a principal fiber bundle whose group is thefirst group of homotopy π1(X). Any other connected covering is aquotient of this one.Simple connexity will be defined in the next lecture.14. Proposition [Universal Covering] Let f : Y → X with X con-nected and Y simply connected. There exists always a smoothlifting f̃ : Y → X̃ , where π : X̃ → X is the universal covering. Thatis, π ◦ f̃ = f . And if we fix three points y ∈ Y, x ∈ X, x̃ ∈ X̃, suchthat f (y) = x, π(x̃) = x, there exists a unique lifting such thatf̃ (y) = x̃.15. Example [The Irrational Tori] If we call irrational tori anyquotient Q of Rn by a discrete subgroup Γ that spans Rn, that is,Γ⊗ R = Rn, then Rn is the universal covering and π1(Q) = Γ.16. Example [Diff(S1)] The universal covering of group of diffeo-morphisms Diff(S1) is naturally identified as

D̃iff(S1) ' {f̃ ∈ Diff(R) | f̃ (x + 1) = f̃ (x) + 1}

Any f̃ ∈ D̃iff(S1) descends on S1 ' R/2πZ by

f (class(x)) = class(f (x)), ou bien f (e2iπx) = e2iπf̃ (x).

Examples of Diffeological Fiber Bundles

17. Example [The Irrational Flow on the Torus] As said previouslythe Projection class : T2 → Tα = T2/S/alpha is a principal fibra-tion with fiber (R, +). That is an interesting example because inordianry differential geometry, the geometry of manifolds: everyfiber bundle with a contractible fiber has a smooth global section.If it is a principal fiber bundle, it is then trivial. That is clearly notthe case for diffeological fiber bundles since T2 is not diffeomorphicto the product Tα × R.18. Example [The Infinite Sphere over the Infinite Projective Space]We recall the construction of the infinite projective space, let

C? = C – {0} and H?C = HC – {0},


where HC is the Hilbert space of infinite square-summable se-quences of complex numbers. We equip that space with the finediffeology of vector space. The plots are the parametrizations thatwrite locally as

P: r 7→∑α∈Aλα(r)ζα,

where A is a finite set of indices, the λ are smooth parametrizationsin C, and the ζα are fixed vector in HCThen, we consider the multiplicative action of the group C? onH?C, defined by

(z, Z) 7→ zZ ∈ H?C, for all (z, Z) ∈ C? × H?C,

and the quotient of H?C by this action of C? is called the infinite

complex projective space, denoted by

PC = H?C/C

?.

Now, HC is equipped with the fine diffeology and H?C with thesubset diffeology. The infinite projective space PC is equippedwith the quotient diffeology. Let

class : H?C → PCbe the canonical projection. For every k = 1, . . . ,∞, let

jk : HC → H?Cbe the injections

j1(Z) = (1, Z) and jk(Z) = (Z1, . . . , Zk–1, 1, Zk, . . .), for k > 1.

Then, let Fk be the map

Fk : HC → PC with Fk = class ◦jk, k = 1, . . . ,∞.That is,

F1(Z) = class(1, Z)

andFk(Z) = class(Z1, . . . , Zk–1, 1, Zk, . . .), for k > 1.

Proposition. The preimage class–1(Val(Fk)) ⊂ H?C is isomorphic tothe product HC × C?, where the action of C? on H?C is transmutedinto the trivial action on the factor HC, and the multiplicativeaction on the factor C?. Therefore, the projection

class : H?C → PC


is a locally trivial C?-principal fibration. We recall that a locallytrivial fibration is stronger than a diffeological fiber bundle whichis trivial only along the plots.

JProof. The subset class–1(Val(Fk)) is the set of Z ∈ H? equiva-lent to some jk(Z′), with Z′ any element in H. That is, Z = zjk(Z′).Let then define

Φk : H × C? → H? by Φk(Z, z) = zjk(Z).

This is a diffeomorphism from H × C? to class–1(Val(Fk)). On theother hand, we know that the Val(Fk), with k ∈ N, is a D-opencovering of PC. Therefore, the projection class : H?C → PC is aprincipal bundle with groups C?. I19. Example [A Remarkable Non Trivial Fiber Bundle] It is wellknown that, in the category of manifolds, a fiber bundle over a con-tractible manifold is trivial. It is also well known that in that cat-egory also, a fiber bundle with contractible fiber admits a smoothsection. Therefore, that are two good reasons for a principal fiberbundle with a contractible group over a contractible manifold tobe trivial.

The following example is an attempt to the construction of a dif-feological fiber bundle over a contractible diffeological space, witha contractible group (that is R), that would be not trivial.We consider the action of Z on C

∀n ∈ Z,∀z ∈ C, n(z) = ze2iπαn,

with α an irrational number. Let

Qα = C/Z

Now, consider the following action of Z on C × R

n(z, t) = (n(z), t+ |z|2n)

Let

C ×Z R = (C × Z)/Z.

We have the following commutative diagram representing this con-struction:


C × R C ×Z R

C × R C/Z

pr1

class

π

classThen,

(1) The space C/Z is contractible.(2) The fiber R of π is contractible.

Question: Is the projection π : C ×Z R → C/Z a R-principal fiberbundle?

References

[Igl85] Patrick Iglesias, Fibrations difféologiques et homotopie, State DoctoralDissertation, Université de Provence (1985), Marseille.

[PIZ13] Patrick Iglesias-Zemmour Diffeology. Mathematical Surveys and Mo-nographs, vol. 185. Am. Math. Soc., Providence RI, (2013).

[IZP20] Patrick Iglesias-Zemmour and Elisa Prato Quasifolds, Diffeology andNoncommutative Geometry. To appear in the Journal of Noncommu-tative Geometry. http://math.huji.ac.il/~piz/documents/QDANCG.pdf

E-mail address: [email protected]

URL, foot: http://math.huji.ac.il/~piz/

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