PDF singular de Rham algebra and spectral sequences in diffeology

A singular de Rham algebra and spectral sequences in diffeology

Katsuhiko Kuribayashi (Shinshu University)

6 June 2021 Global Diffeology Seminar

Online on Zoom

Contents

§1. A stratifold – As a diffeological space –

§2. The singular de Rham complex in diffeology

§3. The de Rham theorem and its applications

– Chen’s iterated integral in Diff, the Leray–Serre spectral sequence and the Eilenberg–Moore spectral sequence –

§4. Future prospective:

With functors aroundDiff and the singular de Rham functor

Diffeological spaces Differential spaces and Stratifolds

§ 1. A stratifold as a diffeological space

The embeddingC^∞( ) :Mfd→R-Alg Definition 1.1 (Sikorski (1971).)

Adifferential spaceis a pair(S,C)consisting of a topological spaceS and anR- subalgebraCof theR-algebraC⁰(S)of continuous real-valued functions onS, which is assumed to belocally detectableandC^∞-closed.

Local detectability : f ∈ C if and only if for anyx ∈ S, there exist an open neighborhoodU ofxand an elementg∈ C such thatf|U =g|U.

C^∞-closedness : For eachn ≥ 1, eachn-tuple(f1, ..., fn)of maps inC and each smooth mapg :Rⁿ → R, the compositeh : S → Rdefined byh(x) = g(f1(x), ...., fn(x)) belongs toC.

Forx∈S,T_xS : the vector space consisting of derivations on theR-algebra Cx

of the germs atx(tangent space).

Diffeological spaces Differential spaces and Stratifolds

Definition 1.2 (Kreck (2010))

Astratifoldis a differential space(S,C)such that the following four conditions hold:

1. S is a locally compact Hausdorff space with countable basis;

2. theskeletaskk(S) :={x∈S |dimTxS ≤k}are closed inS;

3. for eachx ∈ S and open neighborhoodU ofxin S, there exists abump functionatxsubordinate toU

4. thestrataS^k :=sk_k(S)−sk_k−1(S)arek-dimensional smooth manifolds such that restriction alongi : S^k ,→ S induces an isomorphism of stalks i^∗:Cx

∼=

→C^∞(S^k)x for eachx∈S^k.

▶ The ‘cone’ is a stratifold.

▶ Aparametrized stratifold(p-stratifold) is constructed from a stratifold at- taching other finite manifolds with boundaries.

A continuous mapf : (S,C)→(S^′,C^′)is amorphism of stratifoldsif ϕ◦f ∈ C for anyϕ∈ C^′. We denote byStfdthe category of stratifolds.

Diffeological spaces Differential spaces and Stratifolds

Proposition 1.3 (Aoki–K (’17))

There is a functork :Stfd→Diff defined byk(S,C) = (S,D_C)andk(f) = f for a morphismf :S →S^′ of stratifolds, where

DC :=

{

u:U →S

U :open inR^q, q≥0,

ϕ◦u∈ C^∞(U)for any ϕ∈ C }

,

LetM be a manifold and(S,C)a stratifold. Then the functork :Stfd→Diff induces a bijection

k_∗:Hom_Stfd((M, C^∞(M)),(S,C)) →^∼= Hom_Diff((M,DC^∞(M)),(S,DC)).

Mfd fully faithful

j //

ℓ:fully faithful

((Stfd ^k //Diff

D

⊥ //

TopSp

C

oo

Observe that the functork above is nothing but the functorΠin the sense of Batubenge, I-Zemmour, Karshon and Watts (’17).

The de Rham complex due to Souriau The original one due to Chen (1977), Souriau (1980)

§ 2. The de Rham complex due to Souriau

For an open setU ofRⁿ, letD^X(U)be the set of plots withU as the domain andΩ^∗_{de Rham}(U)the usual de Rham complex ofU. LetOpenbe the category consisting of open sets of Euclidian spaces and smooth maps between them.

Ω^p(X) :=





 Open^op

D^X **

Ω^p_{de Rham}

44

^ω Sets

ω is a natural transformation







with the cochain algebra structure induced by that ofΩ^∗_{de Rham}(U).

Remark 2.1

LetM be a manifold andΩ^∗_deRham(M)the usual de Rham complex ofM. Re- call thetautological mapθ : Ω^∗_deRham(M)→Ω^∗(M)defined by

θ(ω) ={p^∗ω}p∈D^M.

Then it follows thatθ is an isomorphism of cochain algebras.

The de Rham complex due to Souriau The original one due to Chen (1977), Souriau (1980)

Iglesias-Zemmour (Canad. J. Math. 65 (2013)) has introduced an integration map of the form ∫ IZ

: Ω^∗(X)−→C_cube^∗ (X) to the cubic cochain complex.

For the irrational torusT_γ²=R/(Z+γZ)，(γ: irrational ) with the quotient diffeology. We see that

Ω^∗(T_γ²)∼= (∧^∗(R¹), d= 0) and thenH¹(Ω(T_γ²))∼=R¹.

On the other hand, by the Hurewicz theorem inDiff enables us to conclude that H¹(C_cube^∗ (T_γ²))∼=R².

▶ One might expect a new de Rham complex for which the ‘de Rham theorem’

holds. (For connecting de Rham calculus and homotopy theory. )

The singular de Rham complex A new de Rham complex

§ 3. The singular de Rham complex

▶ The cubic de Rham complex (Iwase – Izumida ’19)

▶ The singular de Rham complex (K ’20) Aⁿ:={(x0, ..., xn)∈Rⁿ⁺¹ |∑n

i=0xi= 1}: a diff-space with subdiffeology of the manifoldRⁿ⁺¹

Define a simplicial DGA(A^∗_DR)_• as follows.

For eachn≥0,(A^∗_DR)n:= Ω^∗(Aⁿ)and define a simplicial set S_•^D(X) :={{σ:Aⁿ →X |σ : C^∞–map}}n≥0

Moreover, we have a simplicial map

S_•^D(X)→S_•^D(X)_sub:={{σ : ∆ⁿ_sub→X |σ is a C^∞-map}}n≥0

induced by the inclusionj : ∆ⁿ_sub→Aⁿ.

The singular de Rham complex A new de Rham complex

Let∆be the category which has posets [n] :={0,1, ..., n}forn≥0as objects and non-decreasing maps[n]→[m]forn, m≥0as morphisms. By definition, a simplicial set is a contravariant functor from∆toSetsthe category of sets.

A^∗_DR(S_•^D(X)) :=





 ∆^op

S_•^D(X)

**

(A^∗_DR)_•

44 ^ω Sets

ω: a natural transformation







Definition 3.1 (For connecting new de Rham to the original one.) Thefactor mapα: Ω^∗(X)→A^∗_DR(S_•^D(X)) is defined by

α(ω)(σ) :=σ^∗(ω).

Variations of thesingular de Rham complexA^∗_DR(S_•^D(X))are considered.

The singular de Rham complex A new de Rham complex

The simplicial DGA(C_{P L}^∗ )_•:=C^∗(∆[•]), where∆[n] =hom_∆(–,[n])is the standardn-simplicial set.

▶ We define an integration map∫

∆^p : (A^p_DR)_p→Rby

∫

∆^p

ω:=

∫

∆^p

θ⁻¹ω,

whereθ : Ω^∗_deRham(A^p) −→^∼= Ω^∗(A^p)is the tautological map mentioned above.

▶ Define a mor. of simpl. DG modules∫

: (A^∗_DR)_•→(C_{P L}^∗ )_•=C^∗(∆[•]) by

(

∫

γ)(σ) =

∫

∆^p

σ^∗γ,

whereγ ∈(A^p_DR)_n,σ :A^p→ Aⁿ is the affine map induced byσ : [p]→ [n]. Then we have a commutative diagram of simplicial sets

(C_{P L}^∗ )_•

= **

φ //(CP L⊗ADR)^∗_•

mult◦(1⊗∫ )

A^∗_DR•

oo ψ

tt ∫

(C_{P L}^∗ )_•.

The singular de Rham complex Our main theorem

The de Rham theorem in diffeology

Theorem 3.2 (K (2020))

For a diffeological space(X,D^X), one has a homotopy commutative diagram C^∗(S_•^D(X)_sub)

= ((

≃

φ //_{(CP L}^∗ _⊗A^∗_DR)(S•^D_(X))

mult◦(1⊗∫ )

A^∗_DR(S^D_•(X))

≃

oo ψ

an “integration”∫

vv

Ω^∗(X)

α the factor mapoo

∫_IZ

yy

C^∗(S_•^D(X)_sub) C_cube^∗ (X)

≃

oo l

in whichφandψ are quasi-isomorphisms of cochain algebras and the integra- tion map∫

is a morphism of cochain complexes.

Moreover, the factor mapα is a quasi-isomorphism if(X,D^X)is a finite di- mensional smooth CW-complex in the sense of Iwase–Izumida, or stems from a p-stratifold via the functorkmentioned above.

The singular de Rham complex Applications

Chen’s iterated integrals in diffeology

M : a diff-space,ω_i ∈Ω^pi(M)for each1≤i≤kandq:U →M^I a plot of the diff-spaceM^I. gω_iq := (id_U ×t_i)^∗q^∗_♯ω_i, where q_♯ :U ×I→M is the adjoint toqandt_i : ∆^k→I denotes the projection in theith factor.

(

∫

ω1· · ·ωk)q :=

∫

∆^k

g

ω1q ∧ · · · ∧ωgkq. Then by definition, Chen’s iterated integralIthas the form

It(ω0[ω1| · · · |ωk]) =ev^∗(ω0)∧g∆^∗(

∫

ω1· · ·ωk), where∆ :e LM →M^I is the lift of the diagonal mapM →M×M.

Theorem 3.3 (K (2020))

LetM be a simply-connected diff-space,dimHⁱ(ADR(S^D_•(M))) < ∞for eachi≥0. Suppose that the factor map forM is a quasi-isomorphism. Then

α◦It: Ω^∗(M)⊗B(A)→Ω^∗(LM)→A^∗_DR(S_•^D(LM)) is a quasi-isomorphism ofΩ^∗(M)-modules.

The singular de Rham complex Applications

The Leray–Serre spectral sequence in diffeology

Theorem 3.4 (K (2020),A^∗(X) :=A^∗_DR(S^D(X)) )

Letπ :E → M be a smooth map between path-connected diffeological spaces with path-connected fibreL which is

i)a fibration in the sense of Christensen and Wu or

ii)the pullback of the evaluation map(ε0, ε1) : N^I →N ×N for a connected diffeological spaceN along an embeddingf :M →N ×N.

Suppose further that in the caseii)the cohomologyH(A^∗(M))is of finite type.

Then one has the Leary–Serre spectral sequence{LSE_r^∗,∗, dr}converging to H(A^∗(E))as an algebra with an isomorphism

LSE^∗₂^,∗ ∼=H^∗(M,H^∗(L))

of bigraded algebras, whereH^∗(M,H(L))is the cohomology with the local coefficientsH^∗(L) ={H(A^∗(L_c))}c∈S^D₀(M)

The singular de Rham complex Applications

The Eilenberg–Moore spectral sequence in diffeology

Theorem 3.5 (K (2020))

Letπ : E → M be the smooth map as in Theorem 3.4 with the same as- sumption,φ : X → M a smooth map from a connected diffeological spaceX for which the cohomologyH(A^∗(X)) is of finite type andEφthe pullback of π alongφ. Suppose further that M is simply connected in case ofi)andN is simply connected in case ofii). Eφ //

E ^fe //

π

N^I

(ε₀,ε₁)

X _φ //M

f //N×N

Then one has the Eilenberg–Moore spectral sequence{EME_r^∗,∗, d_r}converging toH(A^∗(Eφ)) as an algebra with an isomorphism

EME^∗₂^,∗ ∼=Tor^∗_H(A^,∗ _∗(M))(H(A^∗(X)), H(A^∗(E))) of bigraded algebras.

The singular de Rham complex Applications

On the proofs.

▶ For the case i), Dress’ construction for the Leary-Serre spectral sequence is applicable to our setting.

▶ For the case ii), the spectral sequences are constructed by considering a smooth lifting problemwith an appropriatehomotopypullback.

Definition 3.6 (Christensen–Wu (2014))

A morphismX → Y inDiff is a fibrationif S_•^D(X) → S^D_•(Y)is a (Kan) fibration inSets^∆op.

FACT

▶ Any diffeological bundle (i.e. the pullback for every global plot is trivial ) with fibrant fibre (for example, a diffeological group) is a fibration [C–W].

▶ For a diff-groupGand a subgroupH with the sub-diffeology, the smooth mapG → G/His a diffeological bundle with fibreH [Iglesias-Zemmour].

Then it is a fibration in the sense of C–W.

The singular de Rham complex Applications

Computational examples

T²:={(e^2πix, e^2πiy)|(x, y)∈R²} ⊃Sγ :={(e^2πit, e^2πiγt)|t∈R}, whereγ ∈R\Q. Then theirrational torusTγ is defined by the quotientT²/Sγ

with the quotient diffeology.

In the categoryDiff,Sγ →T²→^π Tγ : a principal diffeological fibre bundle.

By using the Leray–Serre s.s., we have H^∗(A(Tγ)) ^π_∼^∗

=

//H^∗(A(T²)) H^∗_DR(T²)∼= ∧(x1, x2)_∼

= factor mapoo

The singular de Rham complex Applications

Recall the ˇCech-de Rham spectral sequence due to Zemmour:

▶ A first quadrant spectral sequence

ΩE₂^p,q ∼=H^q(HH^p(RM^op,Ω^∗(NX)), dΩ),

ΩE_r^∗,∗=⇒H^∗(TotC^∗,∗)∼=HH^∗(RM^op,map(G,R)) =: ˇH(X)

▶ Comparing the spectral sequences forΩ(X)andA(X), we have a commu- tative diagram

H¹(Ω(X))⊕ΩE₃^{1,0 Θ}_∼

= //H¹(A(NX)^M)⊕AE₃^1,0 Hˇ¹(X;R).

edge₂

∼= 33

edge₂

∼=

jj

In particular, we see

Θ :H¹(Ω(Tγ))⊕ΩE^1,0₂ →^∼= H¹(A(Tγ))

The singular de Rham complex Applications

Corollary 3.7 (K ’21)

There exists an isomorphismH^∗(A(Tγ)) ∼= ∧(Θ(t),Θ(ξ))of algebras, where t ∈ H^∗(Ω(Tγ)) ∼= ∧(t)is a generator andξ ∈ Fl^•(Tγ) ∼= Ris a flow bundle overTγ with a connection1-form, which is a generator of the groupFl^•(Tγ).

▶ Letf :M →Tγ be a smooth map from a diffeological space M. Then via the pullback construction along the mapf, (*) : S_γ →M ×Tγ T² →^π′ M : a principal diffeological bundle

▶ Then the Leray–Serre spectral sequence in Theorem 3.4 for the fibration (*) allows us to deduce that

(π^′)^∗:H^∗(A^∗(M)) ^∼= //H^∗(A^∗(M×T_γ T²)) of algebras, whereA^∗(–) :=A^∗_DR(S_•^D(–)).

The singular de Rham complex Applications

▶ Suppose further thatM is simply connected. Then the comparison of the EMSS’s in Theorem 3.5 forLM andL(M ×Tγ T²)allows us to obtain an algebra isomorphism

(Lπ^′)^∗:H^∗(A^∗(LM)) −→^∼= H^∗(A^∗(L(M ×T_γ T²)).

By Theorem 3.3 (On the compositeα◦It), we have Assertion 3.8

IfH^∗(A^∗(M)) ∼= H^∗(A^∗(S^2k+1))as an algebra withk ≥ 1and the factor map forM is a quasi-isomorphism, then

H^∗(A^∗(L(M ×T_γ T²)))∼=∧(α◦It((π^′)^∗(ω)))⊗R[α◦It(1⊗(π^′)^∗(ω))]

as anH^∗(A^∗(M))-algebra, whereα is the factor map andωdenotes the vol- ume form ofM.

Further prospects Toward rational homotopy theory for non simply-connected diff-spaces

§ 4. With functors and a model structure on Diff

Assertion 4.1 (With the simplicial DGA(A^∗_DR)_•= Ω^∗(A^•)))

Mfd

ℓ: embedding

((

U: forgetful

11

j

embedding //Stfd ^k //Diff

S^D( )

⊥

tt _D⊢

Sets^∆op

| |D

44

| | ** Top

S( )

⊤

jj ^COO

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