On the cohomology algebras of the free loop space of the real projective space and its diﬀeological version

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On the cohomology algebras of the free loop space of the real projective space and its diﬀeological version

Katsuhiko Kuribayashi (Shinshu University)

17 May 2023 Global Diﬀeology Seminar

Online on Zoom arXiv:2301.09827

Katsuhiko Kuribayashi The cohomology algebras ofLRP nand its diﬀeological versionGlobal Diﬀeology Seminar 2023 1 / 23

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§1. A multiplicative spectral sequence for the nerve of a topological category

LetC = [C1 t //

s //C0] be a topological category. The nerve functor gives rise to a cosimplicial cohain complex

n7→C^∗(NervenC,K) =:C^n,^∗

and then this induces a cosimplicial abelian group n7→H^q(NervenC,K) for anyq, where Kis a ﬁeld.

Let BC be the classifying space, namely, BC= ||Nerve_•C||which is the fat geometric realization of the simplicial space Nerve_•C

ω∪T η:= (−1)^qp^′(d^h_p+1· · ·d^h_p+p_′)^∗ω∪(d^h₀· · ·d^h_p₋₁)^∗η forω∈C^p,q andη ∈C^p^′^,q^′.

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▶ A multiplication on TotC^∗(Nerve_•C,K)

Theorem 1.1 LetC = [C1

t //

s //C0] be a category internal toTop. Then there exists a spec-

tral sequence{E_r^∗^,^∗, dr}converging to H^∗(BC;K)as an algebra with E₂^p,q ∼=H^p(H^q(Nerve_•C;K)).

▶ The cohomology of the TotC^•^,^∗ ∼=H^∗(BC)by a method of acyclic models.

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Multiplicative spectral sequences

Theorem 1.2 (Gugenheim–May, ’74: A torsion functor version)

LetGbe a topological group andX aG-space. Then there exists a spectral sequence converging to the Borel cohomology

H_G^∗(X;K) :=H^∗(EG×GX;K)

as an algebra withE₂^p,q∼=H^p(H^q(Nerve_•G;K)).HereG := [G×X

t //

s //X]

denotes the transformation groupoid associated to theG-spaceX. In particular, one has an isomorphism

E₂^p,q ∼=Cotor^p,q_H_∗_(G;_K₎(K, H^∗(X;K)) providedH^∗(G;K)andH^∗(X;K)are locally ﬁnite.

C : a comodule over Ka ﬁeld andM,N : rightC-comodule and a left one, respactively.

M2CN :=Ker ∇ //M⊗N^∇^:=^∇^M^⊗¹⁻¹^⊗∇^N//M ⊗C⊗N

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§2. The computation of the cohomology algebra of the free loop space of the real projective space

LetGbe a discretegroup acting on a topological spaceM. Forg∈ G, we deﬁne Pg(M) :={γ : [0,1]→M |γ(1) =gγ(0)}

which is a subspace of the space of continuous pathsM^[0,1]from the interval [0,1]toM. Moreover, we put

PG(M) := a

g∈G

(Pg(M)× {g}). (1)

Then,PG(M)admits aG-action deﬁned byh·(γ, g) = (_hγ, hgh⁻¹), where

hγ(t) =h·γ(t). For a spaceX, letLX denote the free loop space ofX, namely, the space of continuous maps from the circleS¹ toX.

LetG→M →^p M/Gbe a principalG-bundle.

To (1) in Section 3, page 18

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The free loop space of the real projective space

Proposition 2.1 (cf. Behrend, Ginot, Noohi and Xu ’12)

The mapp:EG×GPG(M)→L(M/G)induced naturally by the projection p:M →M/Gis a weak homotopy equivalence.

Sketch of proof. For eachm∈ M, a ﬁbration PG^m(M) //PG(M)^q:=

`q_g

//M

is constructed, whereP_G^m(M) =`

g∈GP_g^m(M)and

P_g^m(M) :={γ: [0,1]→M |γ(0) =m, γ(1) =gγ(0) =gm}.Moreover, we have a commutative diagram

P_G^m(M)

p //Ω_[m](M/G)

EG×GPG(M)

1×Gev0

p //L(M/G)

ev₀

EG×GM

e π

≃ //M/G

in which two vertical sequences are ﬁbrations. We can provepon the ﬁbre is a weak homotopy equivalence.

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▶ On the cohomology ofPg(M)for a simply connectedG-spaceM For eachg ∈G, in order to compute H^∗(Pg(M);K), we may use the Eilenberg–Moore spectral sequence (henceforth EMSS) for the pullback diagram

Pg(M) //

M^[0,1]

ε0×ε1

M 1×g //M×M,

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whereε_i is the evaluation map atifori= 0,1andgdenotes the map induced by the action onM with the elementg. We observe that the EMSS{E^∗_r^,^∗, dr} converges toH^∗(Pg(M);K)as analgebrawith

E₂^∗^,^∗∼=Tor^∗_H^,^∗_∗_(M_;_K₎_⊗_H_∗_(M;_K₎(H^∗(M;K)g, H^∗(M;K))

as abigraded algebra. HereH^∗(M;K)g is the cohomology algebraH^∗(M;K) endowed with the rightH^∗(M;K)⊗H^∗(M;K)-action deﬁned by

a·(λ⊗λ^′) =a(λg^∗(λ^′)) fora∈H^∗(M;K)g andλ, λ^′∈ H^∗(M;K).

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▶ On theG-action onPG(M).

Forh∈G, theG-action onPG(M)inducesh_∗:Pg(M)→ Phgh⁻¹(M) which ﬁts in the commutative diagram

Pg(M)

h_∗ )) //

M^[0,1]

h_∗

((

ε₀×ε₁

Phgh⁻¹(M) //

M^[0,1]

ε0×ε1

M

h **

1×g //M×M _h_×_h

**M

1×hgh⁻¹

//M ×M.

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Then, the naturality of the EMSS gives rise to a morphism of spectral sequences which is compatible with the map

(h_∗)^∗:H^∗(Phgh−1(M);K)→H^∗(Pg(M);K).

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Theorem 2.2 (K. 2023)

Letpbe an odd prime or 0, then as algebras,

H^∗(LRP^2m+1;Z/p) ∼= H^∗(LS^2m+1;Z/p)⊕H^∗(LS^2m+1;Z/p)

∼= (∧(y)⊗Γ[y])^⊕² and H^∗(LRP^2m;Z/p) ∼= (∧(x⊗u)⊗Γ[w])⊕Z/p,

wheredegy = 2m+1,degy= 2m,deg(x⊗u) = 4m−1,degw= 4m−2 andZ/0 :=Q.

Sketch of Proof. RPⁿ=Sⁿ/(Z/2),G:=Z/2. By applying Theorem 1.2 to the groupoid [G× PG(Sⁿ) ////PG(Sⁿ)], we have a spectral sequence {E^∗_r^,^∗, d_r}converging toH^∗(LRPⁿ;Z/p)with

E₂^∗^,^∗∼=Cotor^∗_H^,^∗_∗_(G)(Z/p, H^∗(PG(Sⁿ))) as an algebra.

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SinceGis abelian, it follows that theGaction onPG(Sⁿ)is restricted to each Pg(Sⁿ)forg ∈G. Then, we see that

L(RPⁿ)'wEG×GPG(Sⁿ) = a

g∈G

EG×GPg(Sⁿ) and

Cotor^∗_H^,^∗_∗_(G)(Z/p, H^∗(PG(Sⁿ))) =⊕g∈GCotor^∗_H^,^∗_∗_(G)(Z/p, H^∗(Pg(Sⁿ))).

We compute the cotorsion functor with the nomalized cobar complex Z/p{τ^∗}^⊗^k⊗H^∗(Pg(Sⁿ)), ∂k =∇G⊗1 + (−1)^k+11⊗ ∇τ∗

k≥0,

whereτ ∈Gdenotes the nontrivial element,

∇τ^∗ :H^∗(Pg(Sⁿ))→Hf⁰(G)⊗H^∗(Pg(Sⁿ)) =Z/p{τ^∗} ⊗H^∗(Pg(Sⁿ)) is the coaction induced by theG-action onPg(Sⁿ)and the projection

G× Pg(Sⁿ)→ Pg(Sⁿ)gives rise to the map∇G. Observe that the complex is nothing but theE₁-term of the spectral sequence {E_r^∗^,^∗, d_r}.

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▶ A generalization of the computation above

LetGbe a ﬁnite group and N a ﬁnite dimensional leftK[G]^∨-comodule. Then the moduleN^∨ is regarded as a leftK[G]-module via the natural map

∇^∨ :K[G]⊗N^∨→N^∨

induced by the left comodule structure∇onN. Thus, by using the isomorphism N ∼= (N^∨)^∨, we considerN a rightK[G]-module.

Lemma 2.3 (cf. Abrams and Weibel, ’02)

Under the setup above, there are isomorphisms of vector spaces

Cotor^∗_K_[G]_∨(K, N)∼=HH^∗(K[G],K⊗N)∼=Ext^∗_K_[G](K,K⊗N) =H^∗(G, N).

Here the moduleN in the Hochschild cohomology is regarded as a rightK[G]- module mentioned above.

▶ Comments

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▶ {E^∗_r^,^∗, dr}: the SS in Theorem 1.2 converging to

H^∗(EG×GPG(M);K)∼=H^∗(L(EG×GM));K).

Corollary 2.4 (cf. Lupercio, Uribe and Xicotencatl, ’08)

LetGbe a finite group acting on a spaceM andPG(M)the space defined in (1). Suppose thatH^∗(PG(M);K)is locally finite and(ch(K),|G|) = 1. Then as an algebra

H^∗(L(EG×GM);K)∼=K2_K[G]^∨H^∗(PG(M);K).

Proposition 2.5 (Benson ’91)

LetGbe a ﬁnite group. Then as an algebra,

H^∗(LBG;K)∼=Cotor^∗_K_[G]_∨(K,(K[G]_ad)^∨),

whereK[G]_addenotes the adjoint representation of G. Especially, ifGis abelian, thenH^∗(LBG;K)∼=H^∗(G;K)^⊕|^G^|as an algebra.

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Since the trivial map v:Pg(M)→ ∗gives rise to aG-equivariant map H0(PG(M);K)→K[G]ad, it follows from Proposition 2.5 that the horizontal edge (p-axis)E₂^∗^,0is a module over the algebra H^∗(LBG;K)via the morphism

v^∗ :H^∗(LBG;K)→Cotor^p,0_K_[G]_∨(K, H^∗(PG(M);K)) of algebras induced byv. Thus, the spectral sequence admits an H^∗(LBG;K)-module structure on the spectral sequence;

•:H^p(LBG;K)⊗E_r^∗^,^∗^v

∗⊗1//E₂^p,0⊗E_r^∗^,^∗^p^r^⊗¹////E_r^p,0⊗E_r^∗^,^∗ ^m //E_r^∗^+p,^∗,

wherepr is the canonical projection andm is the product structure on the Er-term. Thus we see thatdr(a•x) = (−1)^pa•dr(x)for

a∈H^p(LBG;K)andx∈ E^∗_r^,^∗.

▶ Diagram

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A diﬀeological counterpart of a result on the cohomology of the free loop space A new de Rham complex

§3. A diﬀeological counterpart of a result on the

cohomology of the free loop space of a Borel construction

We recall

▶ the singular de Rham complex (K ’20)

▶ the factor map

▶ Chen’s iterated integral map in diﬀeology

∆ⁿ:={(x0, ..., xn)∈Rⁿ⁺¹ |Pn

i=0xi= 1, xi ≥0}: the standard simplex in the sense of Kihara: In particular,

Λⁿ_k :{(x0, ..., xn)∈∆ⁿ|xi= 0for somei6=k},→∆ⁿ has a smooth retraction forn≥1and0≤k≤n.

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: ∆ⁿ→X |σ : C^∞–map}}n≥0

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∆: objects [n] :={0,1, ..., n}forn≥0,

morphisms : non-decreasing maps[n]→[m]forn, m≥ 0

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 ^η Sets

η : a natural transformation







Deﬁnition 3.1 (Connecting the singular de Rham to the original one.)

Thefactor mapα: Ω^∗(X)→A^∗_DR(S_•^D(X)aﬀ)is deﬁned by α(ω)(σ) :=σ^∗(ω),

whereS^D_•(X)aﬀ:={{σ:Aⁿ →X |σ : C^∞–map}}n≥0

The inclusionι: ∆ⁿ→Aⁿ,(ι^∗)^∗ :A^∗_DR(S^D_•(X)) →^≃ A^∗_DR(S^D_• (X)aﬀ).

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M : a diff-space,ω_i ∈Ω^pⁱ(M)for each1≤i≤kandq:U →MÎ a plot of the diff-spaceMÎ. gω_iq := (id_U ×t_i)^∗q^∗_♯ω_i, where q_♯ :U ×R→M is (the composite of a cut-off function and) the adjoint toq, andti:R^k→R denotes the projection in theith factor.

( Z

ω1· · ·ωk)q :=

Z

∆^k

g

ω1q ∧ · · · ∧ωgkq.

Then by deﬁnition, Chen’s iterated integralIthas the form It(ω0[ω1| · · · |ωk]) =ev^∗(ω0)∧g∆^∗(

Z

ω1· · ·ωk),

where∆ :e L^∞M →M^I is the lift of the diagonal map M →M ×M. Theorem 3.2 (K. ’20)

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

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

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LetGbe a ﬁnite group acting freely and smoothly on a manifoldM. Then, we have the principalG-bundleG→M →^p M/Gin the category of manifolds.

LetP_G^∞(M)be the diﬀeological space obtained by applying the construction (1) inDiﬀ. We consider a smooth map

e

p:P_G^∞(M)→L^∞(M/G)

deﬁned byp((γ, g)) =e p◦γ, where L^∞(M/G)denotes the diﬀeological free loop space onM/G.

For a diﬀeological spaceX, we may writeH_DR^∗ (X)for the singular de Rham cohomologyH^∗(ADR(S_•^D(X))).

Theorem 3.3 (K. ’23)

Under the same setting as above, suppose further thatM is simply connected.

Then, the smooth mappegives rise to a well-deﬁned isomorphism e

p^∗ :H_DR^∗ (L^∞(M/G))→^∼⁼ R2_R[G]^∨H_DR^∗ (P_G^∞(M))

of algebras.

From (1) in Section 1, page 6

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▶ The functorD:Diﬀ →Top; (D-topology)

LetM andN be diﬀeological spaces and C^∞(M, N)the space of smooth maps fromM toN with the functional diﬀeology.

Moreover, the inclusion i:D(C^∞(M, N))→C⁰(DM, DN)is contiunuous (Christensen, G. Sinnamon and E. Wu, ’14). Thus, it follows that the functorD induces a morphism

ξ:S_•^D(C^∞(M, N))^{D( )}→ Sing_•(DC⁰(M, N))→ⁱ^∗ Sing_•(C⁰(DM, DN)) of simplicial sets.

Theorem 3.4 (Smoothing theorem, Kihara ’21)

LetM andN be ﬁnite dimensional manifolds. Then, the well-deﬁned mapξ : S_•^D(C^∞(M, N)) →Sing_•(C⁰(DM, DN))is a weak homotopy equivalence.

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For a manifoldM, we consider the composite

λ:=i◦Dj :D(P_g^∞(M)) //D(C^∞([0,1], M)) //C⁰(D[0,1], DM), wherej is the smooth inclusion. SinceDM =M andD[0,1]is the subspaceI ofR, it follows thatλ:D(P_g^∞(M))→ Pg(M)is continuous. Therefore, the composite

ξ^′:=λ_∗◦D( ) :S_•^D(Pg^∞(M)) →Sing_•(Pg(M)) is a morphism of simplicial sets.

Lemma 3.5

LetM be a simply-connected manifold. Then, one has a sequence of quasi- isomorphisms

APL(Sing_•(Pg(M)))⊗QR^(ξ_≃^′⁾^∗//APL(S_•^D(Pg^∞(M)))⊗QR _≃^ζ //ADR(S^D_• (Pg^∞(M))).

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Sketch of Proof. Consider the following commutative diagram.

H^∗(ADR(S_•^D(Pg^∞(M))) Tor^∗_A

DR(S^D(M×2))(ADR(S^D(M)), ADR(S^D(M^I)))

∼= EMoo ₁

H^∗(AP L(S^D_•(Pg^∞(M))))_R

H(ζ)

∼=

OO

Tor^∗_A

P L(S^D(M^×2))(AP L(S^D(M)), AP L(S^D(M^I)))_R

EMoo ₂

Tor(ζ,ζ)

∼=

OO

H^∗(AP L(Sing_•(Pg(M))))_R

H((ξ^′)^∗)

OO

Tor^∗_A

P L(M^×2)(AP L(Sing_•(M)), AP L(Sing_•(M^I))_R

∼= EMoo 3

Tor(ξ^∗,ξ^∗)

OO

▶ EM_i; the Eilenberg-Moore map

EM₃: iso. (the oroginal one due to Eilenberg-Moore), EM₁:iso. (K. ’20)

▶ ζ:AP L(–)⊗_QR→^≃ ADR(–)(K. ’20)

We deﬁne a quasi-isomorphismξf1:=ζ◦(ξ^′)^∗ the composite in Lemma 3.5.

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Sketch of the proof of Theorem 3.3. For the translation groupoid [G× PG^∞(M)

t //

s //PG^∞(M)], we see thats◦pe=t◦p. Then the mape e

p^∗:H_DR^∗ (L^∞(M/G))→H_DR^∗ (P_G^∞(M))induced bypefactors through R2_R[G]^∨H_DR^∗ (P_G^∞(M)). We show that the morphismpe^∗ of algebras in the theorem is an isomorphism. Consider a commutative diagram

H^∗(C⁰(S¹, M/G);R) ^(e^ξ)

∗

∼= //H_DR^∗ (C^∞(S¹, M/G)) H^∗(L(M/G);R) ^(e^ξ)

∗ //

(q^∗)^∗ ∼=

OO

e p^∗ ∼=

H_DR^∗ (L^∞(M/G))

(q^∗)^∗

∼=

OO

e p^∗

R2_R[G]^∨H^∗(PG(M);R)

(fξ1)^∗

//R2_R[G]^∨H^∗_DR(P^∞_G(M))

in which eachξeis the composite of the morphism induced byξ in the smooth theorem and the quasi-isomorphismζ:AP L(K)⊗QR→ADR(K)for a simplicial setKmentioned above.

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A diﬀeological counterpart of a result on the cohomology of the free loop space Applications

Theorem 3.6 (K. ’23) One has sequences

H_DR^∗ (L^∞RP^2m+1) ^p^e

∗

∼= //R2_R[G]∨H_DR^∗ (PG^∞(S^2m+1))

∧(α◦It(v2m+1))⊗R[α◦It([v2m+1)]_⊕2

and

∼=

OO

H_DR^∗ (L^∞RP^2m) ^p^e

∗

∼= //R2_R[G]^∨H_DR^∗ (P_G^∞(S^2m))

∧(α◦It(v2m[v2m]))⊗R[α◦It(1[v2m|v2m])

⊕R

∼=

OO

of isomorphisms of algebras, wherev_n denotes the volume form onH_DR^∗ (Sⁿ), Itandαare Chen’s iterated integral map and the factor map, respectively.

Note: Chen’s iterated integral map does not work for a non-simply connected manifoldM! (when consideringH^∗(LM))

On the cohomology algebras of the free loop space of the real projective space and its diﬀeological version