Classifying Spaces of Diffeological...

Classifying Spaces of Diffeological Groups

Jordan Watts

Central Michigan University

July 30, 2019

Jordan Watts 12th International ISAAC Congress - Universidade de Aveiro

Joint Work

This talk is based on joint work with Jean-Pierre Magnot(Université D’Angers).

Motivation

Fix α ∈ RrQ.Let Z2 act on R by

(m,n) · x = x+m+ nα.

The orbits are dense, and so the orbit space T has trivialtopology.T is a group, but would not typically be considered atopological group.Is there a “differentiable structure” that we could equip Twith?

Motivation

(m,n) · x = x+m+ nα.

Motivation

(m,n) · x = x+m+ nα.

Motivation

(m,n) · x = x+m+ nα.

Motivation

(m,n) · x = x+m+ nα.

Diffeology

DefinitionFix a set X. A parametrisation is a (set-theoretical) mapp : U → X where U is some open subset of some Euclideanspace.

A diffeology D on X is a family of parametrisations, calledplots, satisfying:

(Covering Axiom) Every constant parametrisation is in D,(Locality Axiom) If p : U → X is a parametrisation thatlocally is equal to a plot in D, then p is in D.(Smoothness Axiom) If p : U → X is in D and f : V → U isa smooth map between open subsets of Euclidean spaces,then p f is in D.

Call (X,D) a diffeological space.

Diffeology

DefinitionA map F : (X,DX)→ (Y,DY ) is (diffeologically) smooth if forevery plot p ∈ DX , the composition F p is in DY .

The resulting category of diffeological spaces is a “complete,co-complete quasi-topos”, meaning you can do almost anythingyou would ever want to do in this category without leaving it:subsets, quotients, function spaces, etc.

Diffeology

DefinitionA map F : (X,DX)→ (Y,DY ) is (diffeologically) smooth if forevery plot p ∈ DX , the composition F p is in DY .

The resulting category of diffeological spaces is a “complete,co-complete quasi-topos”, meaning you can do almost anythingyou would ever want to do in this category without leaving it:subsets, quotients, function spaces, etc.

D-Topology

DefinitionLet (X,DX) be a diffeological space. The D-topology on X isthe strongest topology making all of the plots of D continuous.

Examples

ExampleSmooth manifolds, smooth manifolds with boundary, andsmooth manifolds with corners are all full subcategories ofdiffeological spaces.

Effective orbifolds are “almost” a subcategory, in that there is anatural functor from the orbifolds to diffeological spaces that isessentially injective on objects.

Examples

ExampleSmooth manifolds, smooth manifolds with boundary, andsmooth manifolds with corners are all full subcategories ofdiffeological spaces.

Effective orbifolds are “almost” a subcategory, in that there is anatural functor from the orbifolds to diffeological spaces that isessentially injective on objects.

Diffeological Groups

DefinitionA diffeological group G is a diffeological space equipped withsmooth multiplication and smooth inverse maps.

Example

Given a diffeological space (X,DX), the group Diff(X) ofdiffeomorphisms of X is a diffeological group.

Example

Our group T = R/Z2 equipped with the quotient diffeology is adiffeological group, an example of what is called an irrationaltorus (Iglesias-Zemmour), or an infra-circle (Weinstein), or aquasi-torus (Prato).

Example

Prequantisation

Let (M,ω) be a symplectic manifold.

The first step in the programme of geometric quantisation is toform a prequantisation bundle, a circle bundle (or complexline bundle) with connection over M whose curvature is ω.

To do this, one requires that the cohomology class [ω] sit in theimage of H2(M ;Z)→ H2(M ;R); that is, ω is required to beintegral.

Prequantisation

Question: What happens if ω is not integral?

Answer: One can still construct a principal bundle over M ;however, the structure group is no longer a circle, but someirrational torus – which one depends on ω.

It thus makes sense to study such principal bundles, and so inturn it makes sense to study classifying spaces of such groups.

Prequantisation

Milnor’s Construction

Fix a diffeological group G.

DefinitionDefine EG to be the quotient

(ti, gi) ∈ [0, 1]N ×GN∣∣∣ ∑ ti = 1,

only finitely many ti are non-zero/∼

where (ti, gi) ∼ (t′i, g′i) if

1 ti = t′i for all i, and2 for each i, ti = t′i 6= 0 implies gi = g′i.

Denote elements of EG by (tigi).

(ti, gi) ∈ [0, 1]N ×GN∣∣∣ ∑ ti = 1,

DefinitionEG comes equipped with a smooth free action of G: for h ∈ G,

h · (tigi) = (tigih−1).

The resulting quotient BG := EG/G is the classifying spaceof G.

ExampleIf G is a Lie group, then the underlying topological space of BGis (homotopy equivalent to) the usual classifying space of G.

Diffeological Principal Bundles

Fix a diffeological group G and a diffeological space X.

DefinitionA D-numerable principal G-bundle π : P → X is adiffeological space P with a smooth surjection π such that

1 each fibre π−1(x) admits a transitive free action of G,2 there is an open cover Uα of X admitting a subordinate

smooth partition of unity,3 π−1(Uα) is G-equivariantly diffeomorphic to Uα ×G.

Results

Theorem (Magnot-W., Christensen-Wu)

Fix a diffeological group G.1 EG→ BG is a principal G-bundle.2 There is a natural isomorphism between BG(·) and [·, BG],

where BG(X) is the set of all D-numerable principalG-bundles and [X,BG] is the set of smooth homotopyclasses of maps X → BG.

3 EG is smoothly contractible.4 πk(BG) ∼= πk−1(G) for all k > 0, where πk(·) is the kth

smooth homotopy group.5 Given a smooth homomorphism ϕ : G→ H between

diffeological groups, we obtain a smooth mapΦ: BG→ BH.

Results

Connection 1-Forms

DefinitionA connection 1-form ω : TP → TeG of a principal G-bundleP → X is a 1-form taking values in TeG such that

1 ω is G-equivariant with respect to the adjoint action,2 for every y ∈ P and ξ ∈ TeG, we have

ω(ξP |y) = ξ

where ξP is induced by the infinitesimal action.

ExampleThe Maurer-Cartan form of a diffeological group G is aconnection 1-form on G→ ∗.

Connection 1-Forms

ω(ξP |y) = ξ

Connection 1-Forms

ω(ξP |y) = ξ

Connection 1-Forms

ω(ξP |y) = ξ

Results

Theorem (Magnot-W.)

Fix a diffeological group G.1 EG→ BG admits a connection 1-form.2 Any D-numerable principal G-bundle P → X in BG(X)

admits a connection 1-form.3 With mild conditions on G, connection 1-forms induce

unique liftings of paths from X to P .

Results

Theorem (Magnot-W.)

Results

Theorem (Magnot-W.)

Results

Theorem (Magnot-W.)

Back to T

CorollaryIf T is an irrational torus, and ω a connection 1-form onET → BT , then

dω descends to a curvature form Ω on BT ,for any diffeological space X and smooth mapF : X → BT , F ∗Ω is the curvature of F ∗ET → X withconnection 1-form F ∗ω,any D-numerable principal T -bundle over X is uniquelycharacterised by a class in H1(Path(X), T ),if X is also (smoothly) simply-connected then any non-zeroclosed 2-form µ on X admits a principal T -bundle for some(possibly irrational) torus T whose curvature is µ; if thisbundle is D-numerable, then µ = F ∗Ω.

Back to T

Disclaimer!

Shortly after our paper was published, a paper of DanChristensen and Enxin Wu appeared with similar results (minusthe results involving the connection 1-form), in which a differentdiffeology on EG was used.

Consequently, they are able to show that EG→ BG is uniqueup to smooth homotopy equivalence; that is, it is truly universal.

Our results on connections are compatible with the diffeologyused by Christensen-Wu, and so the results mentioned in thistalk are still true when using their diffeology.

Disclaimer!

Thank you!

References

Jean-Pierre Magnot and Jordan Watts, “The diffeology ofMilnor’s classifying space”, Topol. Appl., 232 (2017),189–213.J. Daniel Christensen and Enxin Wu, “Smooth classifyingspaces” (submitted)https://arxiv.org/abs/1709.10517

Classifying Spaces of Diffeological...

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