van est differentiation and van est integration · based on joint work with: david li-bland...

Van Est differentiation and Van Est integration

Eckhard Meinrenken

Global Poisson Seminar, April 2020

Eckhard Meinrenken Van Est differentiation and Van Est integration

Based on joint work with:

David Li-Bland (L’Ens. Math. 61 (2015)),

Maria Amelia Salazar (Math. Ann. 376 (2020)).


Overview

For a Lie group G , there is the group cochain complex

(C•(G ), δ)

where Cp(G ) = C∞(Gp), and

(δf )(g1, . . . , gp+1)

= f (g2, . . . , gp+1) +

p∑i=1

(−1)i f (g1, . . . , gigi+1, . . . , gp+1)

+ (−1)p+1f (g1, . . . , gp).

For a Lie algebra g, there is the Chevalley-Eilenberg complex

(C•(g),d)

where Cp(g) = ∧pg∗, and d = dCE .


Overview

Theorem (Van Est, 1953)

There is a canonical cochain map

VEG : C•(G )→ C•(g)

inducing an isomorphism in low degree (dependingon connectivity properties).

Working with the complex C•(G )e of germs of functions, getisomorphism in all degrees. (Houard, Swierczkowski).


Overview

Let K be a maximal compact subgroup of G .

Theorem (Van Est, 1955)

There is a canonical isomorphism between thecohomology of (C•(G ), δ) and the cohomology of(C•(g)K−basic, d).

Dupont, Shulman-Tischler, Guichardet (1970s) described explicitcochain maps

RG/K : C•(g)K−basic → C•(G )

realizing this isomorphism.


Overview

Let G ⇒ M be a Lie groupoid, with Lie algebroid A = Lie(G ).

Weinstein-Xu (1991) constructed a cochain map

VEG : C•(G )→ C•(A).

Crainic (2003) proved versions of van Est’s theorems in thiscontext.

Li-Bland-M (2015) gave construction of VEG , and of a cochainmap C•(A)→ C•(G )M , using ‘Perturbation Lemma’.

Cabrera-Marcut-Salazar (2017) gave a formula for a right inverse

RG : C•(A)→ C•(G )M .


Overview

Objectives:

Conceptual understanding of these maps and their properties;avoiding ‘calculations’.

Show that maps of Cabrera-Marcut-Salazar and Li-Bland-Mare the same.

Further generalizations, new results.


I. The Perturbation Lemma


The Perturbation Lemma: setting

Given a double complex

D0,1δ//

d

OO

D1,1δ//

d

OO

D0,0δ//

d

OO

D1,0δ//

d

OO

we have corresponding total complex

(Tot•(D),d + δ).


The Perturbation Lemma: setting

Consider a double complex with augmentations

X1i//

d

OO

D0,1δ//

d

OO

D1,1δ//

d

OO

X0i//

d

OO

D0,0δ//

d

OO

D1,0δ//

d

OO

Y0δ//

j

OO

Y1δ//

j

OO

with a horizontal homotopy h: D•,• → D•−1,•

hδ + δh = idD −i p, p: D0,• → X•, p i = idX .

Claim: can turn h into homotopy operator for (Tot•(D),d + δ).


The Perturbation Lemma: statement

Lemma (Brown, Gugenheim)

Suppose[h, δ] = idD−i p, p i = idX .

Put h′ = h(1 + dh)−1, p′ = p(1 + dh)−1. Then

[h′, d + δ] = idD−i p′, p′ i = idX .

(Proof: Direct verification.) So, can invert the second map in

Y•j−→ Tot•(D)

i←− X•,

thereby producing a cochain map

p′ j = p (1 + dh)−1 j : Y• → X•.


The Perturbation Lemma: statement

On elements of degree p, the map

p (1 + dh)−1 j = p (−dh)p j : Yp → Xp

is a ‘zig-zag’:

X2 D0,2poo

D0,1

d

OO

D1,1−hoo

D1,0

d

OO

D2,0−hoo

Y2

j

OO


The Perturbation Lemma: example

Example (Cech-de Rham double complex)

Let M be manifold with open cover U = Ua.

Cp(U ,Ωq) =⊕

Ωq(Ua0 ∩ · · · ∩ Uap)

Ω1(M)

i//

d

OO

C 0(U ,Ω1)δ//

d

OO

C 1(U ,Ω1)δ//

d

OO

Ω0(M)

i//

d

OO

C 0(U ,Ω0)δ//

d

OO

C 1(U ,Ω0)

d

OO

δ//

C 0(U ,R)?j

OO

δ// C 1(U ,R)

?j

OO

δ//



Example (Cech-de Rham double complex)

Partition of unity χa horizontal homotopy

(hω)a0···ap−1 =∑a

χa ωa a0···ap−1 , [h, δ] = 1− i p

where p: ωa 7→∑

a χaωa.So, can ‘invert’ the second map in

C •(U ,R)j−→ Tot•(C (U ,Ω))

i←− Ω•(M)

to produce a Cech-de Rham cochain map

C •(U ,R)j−→ Tot•(C (U ,Ω))

p(1+dh)−1

−→ Ω•(M)

For good cover this is a quasi-isomorphism. (Homotopy inverse toj constructed by Perturbation Lemma as well).



Cf. Bott-Tu, Differential Forms in Algebraic Topology, page 102:


The Perturbation Lemma: addendum

Suppose we also have vertical homotopy k, so [d, k] = idD−j q.So, have zig-zags in both directions:

φ• = p (1 + dh)−1 j : Y• → X•

ψ• = q (1 + dk)−1 i : X• → Y•.

Lemma

Suppose h k = 0, p k = 0. Then φ• ψ• = idX.

Idea of proof: successively ‘shorten’ the zig-zags.


X2i //

D0,2p

oo

−k

D0,1

d

OO

δ //D1,1

−hoo

−k

D1,0

d

OO

δ //D2,0

−hoo

q

Y2

j

OO


X2i //

D0,2p

oo

−k

D0,1

d

OO

δ //D1,1

−hoo

−k

D1,0

d

OO

δ //D2,0

−hoo

Y2


X2i //

D0,2p

oo

−k

D0,1

d

OO

δ //D1,1

−hoo

−k

D1,0

d

OO

D2,0

Y2


X2i //

D0,2p

oo

−k

D0,1

d

OO

δ //D1,1

−hoo

D1,0 D2,0

Y2


X2i //

D0,2p

oo

−k

D0,1

d

OO

D1,1

D1,0 D2,0

Y2


X2i //

D0,2p

oo

D0,1 D1,1

D1,0 D2,0

Y2


X2

idX

D0,2

D0,1 D1,1

D1,0 D2,0

Y2

Q.E.D.


II. Van Est maps for Lie groups and Lie algebras


Van Est double complex

Recall Cp(G ) = C∞(Gp), Cq(g) = ∧qg∗.

Van Est double complex (D(G ), δ,d):

Dp,q(G ) = C∞(Gp × G ,∧qg∗),

with

(δψ)(g1, . . . , gp+1; g)

= ψ(g2, . . . , gp+1; g) +

p∑i=1

(−1)iψ(g1, . . . , gigi+1, . . . , gp+1; g)

+ (−1)p+1ψ(g1, . . . , gp; gp+1g)

and d = (−1)pdCE (using principal G -action on last factor).

Horizontal and vertical augmentation maps:

i : C•(g)→ D0,•(G ), j : C•(G )→ D•,0(G ).



∧1g∗

i//

d

OO

C∞(G 0 × G ,∧1g∗)δ//

d

OO

C∞(G 1 × G ,∧1g∗)δ

//

d

OO

∧0g∗

i//

d

OO

C∞(G 0 × G ,∧0g∗)δ//

d

OO

C∞(G 0 × G ,∧1g∗)

d

OO

δ//

C∞(G 0)?

j

OO

δ// C∞(G 1)

?

j

OO

δ//

For K ⊆ G , this restricts to the K -basic subcomplexes

D•,•(G )K−basic, (∧•g∗)K−basic.


Van Est differentiation

HereDp,q(G )K−basic = C∞(Gp × G , (∧qg∗)k−hor)

K .

Lemma

For K ⊆ G compact (e.g. K = e) have horizontal homotopyoperator,

h = AvK hG

where AvK denotes averaging, and

(hGψ)(g1, . . . , gp−1; g) = (−1)pψ(g1, . . . , gp−1, g ; e)

So, we obtain a cochain map

VEG/K = p (1 + dh)−1 j : C•(G )→ C•(g)K−basic.

Q: What is this map?Eckhard Meinrenken Van Est differentiation and Van Est integration

Van Est differentiation

Consider Gp+1-action on Gp

(a0, . . . , ap) · (g1, . . . , gp) = (a0g1a−11 , a1g2a

−12 . . . , ap−1gpa

−1p ).

Let AvK : C∞(Gp)→ C∞(Gp) averaging wrt Kp+1 ⊆ Gp+1.

Theorem (M-Salazar)

We have VEG/K = VEG AvK with

VEG (f ) = (d(1) · · · d(p)f )∣∣(e,...,e)

.

Hered (i) : C∞(Gp,∧g∗)→ C∞(Gp,∧g∗)

is CE differential for diagonal action of G ⊆ Gp+1−i ⊆ Gp+1.

VEG is the ‘standard’ van Est map.


Van Est integration

For integration, we identify

(∧qg∗)K−basic = Ωq(G/K )G .

Similarly,Dp,q(G )K−basic = C∞(Gp, Ωq(G/K )).

Here, dCE becomes de Rham differential.

Suppose K ⊆ G is maximal compact. Then

G/K ∼= g/k.

For G semisimple, follows from Cartan decomposition: G = K × p.


Van Est integration

The linear retraction of G/K ∼= g/k defines a de Rham homotopyoperator,

k : C∞(Gp, Ωq(G/K ))→ C∞(Gp, Ωq−1(G/K ))

and hence a van Est integration map

RG/K = q (1 + δk)−1 i : C•(g)K−basic → C•(G ).

Theorem (M-Salazar)

The van Est integration map RG/K is a right inverse to the van Estdifferentiation map VEG/K .

Proof: check h k = 0, p k = 0.

Q: What is the map RG/K?


Van Est integration

(∧g∗)K−basic∼= Ω(G/K )G , α 7→ αG/K .

Theorem (M-Salazar)

The cochain map RG/K : (∧g∗)K−basic → C∞(Gp) is given by

RG/K (α)(g1, . . . , gp) =

∫[0,1]p

γ(g1, . . . , gp)∗αG/K

where γ(g1, . . . , gp) : [0, 1]p → G/K defined below.

Definition ofγ(g1, . . . , gp)

∣∣t1,··· ,tp

∈ G/K

uses retraction λt of G/K ∼= g/k:

eK → G/Kgp−→ G/K

λtp−→ G/K · · · g1−→ G/Kλt1−→ G/K


III. Lie groupoids and Lie algebroids


Lie groupoids

We’ll now consider similar techniques for Lie groupoids G ⇒ M.

G manifold of arrows;M ⊆ G submanifolds of units;s, t : G → M source, target (surjective submersions):

t(g) s(g)

g

Multiplication = concatenation (gh, defined when s(g) = t(h))

Units = trivial arrows

Inversion = reversing arrows


Lie groupoids

The space BpG ⊆ Gp of p-arrows

m0 m1 m2 · · · · · · mp

g1 g2 g3gp

is a simplicial manifold, with face maps

∂i : BpG → Bp−1G

dropping mi . Explicitly,

∂i (g1, . . . , gp) =

(g2, . . . , gp) i = 0

(g1, . . . , gigi+1, . . . , gp) 0 < i < p

(g1, . . . , gp−1) i = p


Lie groupoids

Groupoid cochain complex: C•(G ) = C∞(B•G ) with differential

δ =

p+1∑i=0

(−1)i∂∗i : Cp(G )→ Cp+1(G )

Variations:

Normalized subcomplex C•(G ): pullbacks under degeneracymaps are 0.

Localized complex Cp(G )M = Cp(G )/ ∼ of smooth germsalong M ⊆ BpG .


Lie groupoids

A Lie groupoid G ⇒ M with source, target maps s, t : G → M hasan associated Lie algebroid A⇒ M:

A = Lie(G ) = TG |M/TM

anchor a : Lie(G )→ TM induced from T t−T s : TG → TM,

[·, ·] from Γ(Lie(G )) ∼= X(G )L

For Lie algebroid A→ M, have Lie algebroid complex

C•(A) = Γ(∧•A∗)

with Chevellay-Eilenberg (or de Rham) differential.


Lie groupoids

Example (Alexander-Spanier complex)

Pair(M)⇒ M pair groupoid Bk Pair(M) = Mk+1. So,

C•(Pair(M)) = C∞(M•+1),

with

(δf )(m0, . . . ,mp+1) =∑

(−1)i f (m0, . . . , mi , . . . ,mp+1)

has trivial cohomology. But localized complex C•(Pair(M))Mcomputes de Rham cohomology.

Actually, for every proper Lie groupoid G ⇒ M the complex C•(G )has trivial cohomology (M. Crainic, 2003).



Given G ⇒ M, define

EpG = (a0, . . . , ap) ∈ Gp+1| s(a0) = . . . = s(ap).

This is a principal G -bundle

EpG

πp

κp// BpG

M

πp(a0, . . . , ap) = the common source,

κp(a0, . . . , ap) = (a0a−11 , a1a

−12 , . . . , ap−1a

−1p )


Van Est double complex for Lie groupoids

Get simplicial manifold

EpG

πp

κp// BpG

M

with face maps ∂i : EpG → Ep−1G deleting i-th entry. Fibration

κp : EpG → BpG fiberwise forms (Ω•F (EpG ), dRh).

Definition (Crainic, 2003)

The double complex (D•,•, δ,d) with

Dp,q = ΩqF (EpG ),

δ =∑

(−1)i∂∗i , d = (−1)pdRh is the van Est double complex.


Van Est double complex for Lie groupoids

Γ(∧1A∗)

i//

d

OO

Ω1F (E0G )

δ//

d

OO

Ω1F (E1G )

δ//

d

OO

Γ(∧0A∗)

i//

d

OO

Ω0F (E0G )

δ//

d

OO

Ω0F (E1G )

d

OO

δ//

C∞(B0G )?

j

OO

δ// C∞(B1G )

?

j

OO

δ//

Here:

i = inclusion as left-invariant (leafwise) forms,

j = κ∗p


Van Est differentiation for Lie groupoids

G. Segal: Classifying spaces and spectral sequences (1968) ⇒

Theorem

There is a canonical simplicial deformation retraction from E•Gonto M.

As a consequence, get a horizontal homotopy operator,

δh + hδ = id−i p, p i = id .

Here p: ΩqF (E0G )→ Γ(∧qA∗) is restriction to M ⊆ E0G = G .

we obtain a van Est differentiation map

VEG : p (1 + dh)−1 j : C•(G )→ C•(A).

Q: What is this map?



Theorem (Li-Bland, M)

The cochain map

VEG : p (1 + dh)−1 j : C•(G )→ C•(A).

is the Van Est map of Weinstein-Xu (1991).

Weinstein-Xu’s map VEG : C∞(BpG )→ Γ(∧pA∗) is given by

VEG (f )(X1, . . . ,Xp) =∑s∈Sp

sign(s) L(X 1,]s(1)) · · · L(X p,]

s(p)) f∣∣∣M

Here X i ,], X ∈ Γ(A), i = 0, . . . , p is the generating vector field forthe i-th G -action on BpG :

h.(g1, . . . , gp) = (g1, . . . , gih−1, hgi+1, . . . , gp).



Example

For the pair groupoid Pair(M)⇒ M, the map

VEPair(M) : C∞(Mp+1)→ Ωp(M)

is given by

VEPair(M)(f0 ⊗ · · · ⊗ fp) = f0 df1 ∧ · · · ∧ dfp


Van Est integration for Lie groupoids

Next, let’s consider vertical homotopies.

Definition (Cabrera-Marcut-Salazar)

A tubular structure for G ⇒ M is a tubular neighborhoodembedding

A = Lie(G )→ G

taking vector bundle fibers to t-fibers.

We’ll pretend that this is a global diffeomorphism.



∀p, the principal G -bundle has a canonical trivialization

EpG ∼= BpG ×M G .

Tubular structure retraction of EpG onto BpG .

vertical homotopy operator k : Ω•F (E•G )→ Ω•−1F (E•G ),

[d, k] = id−j q;

here q is given by the pullbacks under BpG → EpG .

we obtain a van Est integration map

RG = q (1 + δk)−1 i : C•(A)→ C•(G )M .

As before, we can check VEG RG = idC(A).



Q: What is this cochain map RG = q (1 + δk)−1 i?

Theorem (M-Salazar)

This composition is the van Est integration map RG ofCabrera-Marcut-Salazar (2017).

Explicitly, with Γ(∧A∗) ∼= ΩF (G )L,

(RGα)(g1, . . . , gp) =

∫[0,1]p

γ(g1, . . . , gp)∗αL.

Hereγ(g1, . . . , gp) : [0, 1]p → G is given by

s(gp) ∈ Ggp−→ G

λtp−→ G · · · g1−→ Gλt1−→ G

(In C-M-S 2017, it was checked by direct calculation that RG iscochain map, right inverse to VEG .)


Further remarks, work in progress

We also consider van Est integrations for proper groupoidactions of G ⇒ M on manifolds Q → M. (Cf. Crainic 2003)

More complicated discussion for more general van Est map ofMehta (2006) and Abad-Crainic (2011),

VEG : Ω•(B•G )→W•,•(A)

In joint work (in progress) with Jeff Pike, we plan togeneralize further to double groupoids, LA-groupoids, doubleLie algebroids and such.


Thanks, and stay safe!


van est differentiation and van est integration · based on joint work with: david li-bland...

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