Higher string topology of product manifolds

since pev₀Ψqpp, zq p meaning thatev₀Ψis a submersion.

We clearly have

Dl rpSⁿSⁿ¹,Ψ, Dp1qqs rpSⁿSⁿ¹,Ψ,0qs 0 . It remains to check that aa0 and that alla.

Recall that for de Rham chains we have

rpU, φ, π_!ωqs rpU¹, φπ, ωqs,

where π_!: Ω^r_cpU¹q ÑΩ^r_c^dim^U¹ ^{dim U}pUq is the integration along the ber dened for C⁸-submersionsπ :U¹ÑU.

We use the concatenationc:c0,0 dened in the previous section. This yields

aacpaSⁿaq c rpDⁿ,Φ, ωqs Sⁿ rpDⁿ,Φ, ωqs c rpDⁿSⁿDⁿ,ΦΦ, πω^πωqs

rpDⁿ,Φ, ω^ωqs 0 for π :DⁿSⁿDⁿÑDⁿ since ω² PΩ²ⁿ_c pDⁿq vanishes.

We further have allasince alc aSⁿl

c rpDⁿ,Φ, ωqs SⁿrpSⁿSⁿ¹,Ψ,1qs c rpDⁿSⁿpSⁿSⁿ¹q,ΦΨ, πω^π1qs c rppSⁿSⁿ¹q SⁿDⁿ,ΨΦ, π1^πωqs la

for the dieomorphismDⁿSⁿpSⁿSⁿ¹q Ñ pSⁿSⁿ¹q SⁿDⁿthat in particular is a C⁸-submersion. FurthercpaSⁿlq cplSⁿaq sinceais a family of constant Moore loops.

For even dimensional spheres this construction does not work since we do not nd a nowhere vanishing vector eld on these spheres. The existence of poles complicates the denition of l as a regular chain.

can be realized by choosing a representative γ_α :S¹ ÑN and considering

Γα :S¹ R{ZÑLN

t Ñγ_αp 1{ltq . forl being the winding number of γα. For α0we have

Γ₀ :N ÑLN pÑγ_p

whereγ_pptq p for all tP S¹. We set imΓ_α :S_α¹ and imΓ₀ :N. ForX :M N the free loop space LX thus topologically looks like

LpM Nq

LM LN LM §

αPπ0pLNq

L^αN Cor. p2.15q

LM pN \ §

0α

P rπ1pNq rπ1pMNq

S_α¹q pLMNq \ pLM §

α0

S_α¹q .

Our goal is to transfer the structure dened above, namely the dg algebra structure and the dg Lie structure of

pCpXq, D,,t,uq to anA₈-algebra and an L₈-algebra on homology

HpCpXq, Dq HpLXq.

The basic idea of the following construction is that we want the subspaces S_α¹ to be disjoint implying that theA₈{L₈-algebra operations on homology are essentially zero.

The construction further yields an A₈-algebra morphism f and an L₈-algebra mor-phismφ that are 8-quasi-isomorphism

HpLXq,tm_nun¥1,tλ_nun¥1 ftfnu // CpLXq,mr₁,mr₂ . and

HpLXq,tm_nun¥1,tλ_nun¥1

φtφnu // CpLXq,λr₁,rλ₂ . where

m₁ :D, mr₂pa, bq: p1q^|^a^|ab , rλ1 :D, rλ2pa, bq: p1q^|^a^|ta, bu .

We work with the disk of radius r dened as Dⁿprq : tx P Rⁿ| |x| ¤ ru and set Dⁿ:Dⁿp1q in the following.

Proposition 4.11

Let A rπ₁pNq be the set of primitive nontrivial homotopy classes of loops in N. Then there exist curves γ_a in N indexed by a P A and closed tubular neighbour-hoods O_a γ_a with the following properties:

piq The curveγa represents the homotopy classa.

piiq O_a is a smooth submanifold of N with boundary and is dieomorphic to S¹Dⁿ¹ via a dieomorphism φ_a:S¹Dⁿ¹ ÑO_a.

piiiq For ab the submanifolds O_a and O_b are disjoint.

Remark that in particular O_a and O_a are disjoint.

Proof : For a0 the curveγ_aN is chosen as a representative ofa.

The manifold N is compact and thus π₀pLNq rπ₁pNq is countable. We choose a counting

A ta₁, a₂, ...u .

Fix γ_a₁ and the closed tubular neighbourhood O_a₁ in N, which is possible due to corollary 2.3 of [23] for example. We have a dieomorphism φa1 :S¹Dⁿ¹ ÑO_a₁. We recursively isotope γ_a_i for i1 and use the same notation for the perturbedγ_a_i. Since we use isotopies the perturbed γai is still a representative of ai.

For the inductive step assume that we have modied γa1, ..., γak and constructed disjoint closed neighbourhoods O_a₁, ...,O_a_k satisfying piq - piiiq of the proposition.

Isotopeγ_a_k ₁ such that

γ_a_k ₁&γ_a₁ , γ_a_k ₁&γ_a₂, ..., γ_a_k ₁&γ_a_k .

Such isotopies exist due Corollary IV.2.4 of [23] for example and the fact that the γai's are smooth compact submanifolds. Since the curvesγai are one dimensional and we assume N to be of dimensionm¥3this implies

γ_a_iXγ_a_j H for0 i, j¤k 1withij .

By radially moving out we can achieve that γa_k ₁ intersectsO_a_i (1¤i¤k) only in BO_a_i :B_a_i S¹Sⁿ² and that

γak 1 XO_a_i H .

These submanifoldsB_a_i, B_a_j N are disjoint, closed and compact, thus have positive pairwise distances d_i,j ¡0. We x disjoint open neighbourhoodsU_a_i ofB_a_i inM for 1¤i¤k. TheBai's are dieomorphic to S¹Sⁿ² and in particular hypersurfaces.

We can achieve that

γ_a_k ₁ XO_a_i H for 1¤i¤k

by perturbing γ_a_k ₁ in U_a_i. After these perturbations for all 1 ¤i¤k the subman-ifolds γa_k ₁ and O_a_i M are disjoint and have a distance di ¡ 0. We thus can

constructO_a_k ₁ as a closed tubular neighbourhood of γ_a_k ₁, and in particular we can arrange

O_a_iXO_a_j H for0 i, j¤k 1withij . This concludes the inductive step and thus proves the proposition.

Remark 4.12. We x a smooth homotopy

H :S¹Dⁿ¹ r0,1s ÑS¹Dⁿ¹ pτ, x, tq ÞÑH_tpτ, xq where H_tpτ, xq is the ow of the vector eld

Vpτ, xq:ρp|x|q _B_x^B_n₁

on S¹ Dⁿ¹ at time t. Here x px₁, ..., x_n₁q P Dⁿ¹ Rⁿ¹ and ρ smooth is a cut-o function of the form

The homotopy H satises

piq H_t id near BpS¹ Dⁿ¹q for all t . piiq H₀ id .

piiiq H1 S¹Dⁿ¹p1{4q

X S¹Dⁿ¹p1{4q

H .

Due to the work of Irie in [20] we know that the homology of the complex CpLpM Nqq

is isomorphic toHpLpMNqq. Further the de Rham loop product and the de Rham loop bracket descend to homology and there they coincide with the loop product and the loop bracket respectively dened by Chas and Sullivan in [5].

We have

CpLpM Nqq C⁰pLpM Nqq `à

aPA

C^apLpM Nqq

where C^apLpM Nqq contains chains in homotopy classes which are positive iterates of a and C⁰pLpM Nqq contains chains of contractible loops.

Remark the subcomplex C¹ CpLpMNqq that splits as

C¹ : pC¹q⁰`à

aPA

pC¹q^a

where pC¹q^a C^apLpM Nqq contains all the chains whose loops are in M φ_apS¹Dⁿ¹q

and pC¹q⁰ C⁰pLpM Nqq contains all the chains whose loops are contractible in M N and further constant in N.

Lemma 4.13

The inclusion of the chain complex

C¹ ãÑCpLpM Nqq induces an isomorphism on homology. In particular

HpC¹q HpLpM Nqq.

FurtherC¹ is closed under the de Rham loop product and the de Rham loop bracket dened in [20].

Proof : By proposition 4.11 for aPA andαkawe have homotopy equivalences

LM §

k¥1

L^kpS¹Dⁿ¹q^idÝÑ^Lφ^aLM §

k¥1

L^kaN

and clearly

LMN ÝÑLML⁰N .

The two complexes C¹ and CpLpM Nqq are the complexes of de Rham chains on the loop spaces on the left and the right respectively. By corollary 4.6 we thus get that the homology of these spaces is isomorphic.

By denition the homotopy equivalences are compatible with the de Rham loop prod-uct and the de Rham loop bracket.

Remark 4.14. The lemma in general holds for Dⁿ¹prq with 0 r ¤1. For reasons of clarity in the upcoming proofs we highlight the radius as C¹_,r if r 1.

The homotopy of remark 4.12 yields chain maps h: à

aPA

pC¹q^a Ñà

aPA

pC¹q^a of degree 1 induced by H and

T : à

aPA

pC¹q^aÑ à

aPA

pC¹q^a

of degree 0 induced by H₁. Further H₀ induces the identity on C¹. These relate to

Dh hD id T (4.23) by proposition 2.5. of [20] which guarantees that smoothly homotopic maps induce chain homotopic ones onC¹.

The topological rewriting and simplication of the set-up will imply that xy0 and tx, yu 0

forx, y P HpLXq being homology classes of loops in non-trivial conjugacy class com-ponents of rπ₁pM Nq since the classes x, y can be either represented as families of loops that are disjoint inN due to (iii) of remark 4.12 and (4.23).

The following theorems state the generalization of this fact to the higher A₈{L₈ -algebra operationsmk¥3 and λk¥3 on homology HpLXq.

Remark that in the following we work with

HpLXq HpL⁰Xq ` à

aPA

HpL^aXq whereHpL^aXq À

αka fork¥1

HpL^αXq when setting

L^aX : §

αka fork¥1

L^αX

forα P rπ₁pXq rπ₁pNq.

In the following theorems we assume X M N and M, N to be smooth, closed and oriented Riemannian manifolds of nite dimension dimM m ¥ 0 respectively dimN n ¥3. Further M is simply connected and N has negative sectional curva-ture.

Theorem 4.15

The homotopy transfer construction for

HpLXq ÝÑCpLXq

equips HpLXq with an A₈-algebra structure pHpLXq,tmkuk¥1q and yields an A₈-algebra morphism

f tf_kuk¥1 :HpLXq ÝÑCpLXq such that:

(i) m₁ 0,

(ii) f₁ is a cycle choosing homomorphism and in particular a quasi-isomorphism, (iii) m₂ corresponds to the loop product, and

(iv) m_kpx₁, ..., x_kq 0

for k ¥ 2 whenever the inputs x_i are classes of families of loops that are non-contractible, that is xi P À

aPAHpL^aXq.

Theorem 4.16

The homotopy transfer construction for

HpLXq ÝÑCpLXq

equips HpLXq with an L₈-algebra structure pHpLXq,tλ_kuk¥1q and yields an L₈-algebra morphism

φ tφ_kuk¥1 :HpLXq ÝÑCpLXq such that:

(i) λ₁ 0,

(ii) φ₁ is a cycle choosing homomorphism and in particular a quasi-isomorphism, (iii) λ₂ corresponds to the loop bracket, and

(iv) λ_kpy₁, ..., y_kq 0

for k ¥ 2 whenever the inputs are elements yi P HpL^aⁱXq for primitive classes a_i PA which are not all equal.

For proving the theorems we apply the homotopy transfer construction presented by Kadeishvili in [21] by recursively constructing the higher operations and morphisms.

Proof of theorem 4.15 : We use the notation from section 3.

For the rst operations we set

m1 0U1 and f1 ι

whereι:HpL^aXq Ñ pC¹_,1_{₄q^afor aPAand ι:HpL⁰Xq Ñ pC¹q⁰ are cycle choosing homomorphism. Thus equation (3.3) is satised, namely

U₁f₁m₁0Dι rm₁f₁ .

The operation m2 r rm2i^b²s onHpLXq is the loop product up to sign due to how r

m₂ : de Rham loop product are constructed by Irie.

It remains to prove pivq. For generalk¥2 we have

U_kpx₁, ..., x_kq

k¸1 s1

m₂pf_spx₁, ..., x_sq, f_k_spx_s ₁, ..., x_kqq

k¸2 i0

k¸1 j2

p1qⁱ ¹ ^|^x¹^| ^... ^|^xⁱ^|f_k_j ₁px₁, ..., x_i, m_jpx_i ₁, ..., x_{i j}q, x_{i j} ₁, ..., x_kq .

We will show that there exist maps

fk:HpLXq^b^k ÝÑC¹ such that

Df_k rm₁f_kU_kf₁m_kU_k

when acting on inputsxithat are classes of families of loops that are non-contractible.

This then yields

m_kpx1, ..., x_kq rU_kpx1, ..., x_kqs 0 for such inputs.

For the induction assume the stated assertion holds up to degree k. We perform the inductive step for kÑk 1.

Assume all operations and morphisms are constructed up to degreek. In the induction hypothesis we assume that the image of f_kpx1, ..., x_kq is contained in the support of f₁px_k ₁q when acting onx₁, ..., x_k ₁as in the condition for pivq. In particular we thus have

m2pfkpx1, ..., xkq, T f1pxk 1qq 0. (4.24) According to the denition of the loop product by Irie we know that for chainsxPC¹ and x_i as in the condition for piiiq we have that the supports of

m₂pf₁px_iq, xq and rm₂pT f₁px_iq, xq

are contained in the support of f₁px_iq and T f₁px_iq respectively. By piiiq of remark 4.12 we thus have

m₂pf₁px_iq, T f₁px_iqq 0 . (4.25) We dene

fk 1 : p1q^d¹ ^{... d}^k ^k¹mr2 pfkbhf1q (4.26) when acting on Hd1pL^aⁱ¹Xq b...bHdk 1pL^a^ik ¹Xq fora1, ..., ai_k ₁ PA. Remark that for x₁, ..., x_k ₂ as in the condition for pivq we get that the image off_k ₁px₁, ..., x_k ₁q is contained in the support of f₁px_k ₂q

Due to he work of Kadeishvili in [21] we can dene f_k ₁ for the remaining cases if at least one input is ofHpL⁰Xq such thatDf_k ₁U_k ₁f₁m_k ₁ implyingm_k ₁ not necessarily zero. We do not want to prove something about these operations here.

It remains to show that

Df_k ₁ U_k ₁

when acting onx₁, ..., x_k ₁ as in the condition for piiiq. Recall the A₈-operations in the case rm_k0 for k¥3, namely

m₁ rm₁0, r

m1p rm2px, yqq mr2p rm1pxq, yq p1q^|^x^| ¹mr2px,mr1pyqq 0, r

m2p rm2px, yq, zq p1q^|^x^| ¹mr2px,mr2py, zqq 0 . Since |f_k| k1 we get

p1q^|^x¹^{| |}^x^k^| ^k¹pDf_k ₁qpx1, ..., x_k ₁q (4.27) D rm2 pfkbhf1q

px1, ..., xk 1q

rm₂ ppDf_kq bhf₁q p1q^|^x¹^{| |}^x^k^| ^kmr₂ pf_kb pDhf₁qq

px₁, ..., x_k ₁q. For the rst summand we use the induction hypothesis

Dfj Uj implying mj 0 for 1¤j¤k. In particular we thus get

DfkUk

k¸1 s1

m2pfsbfksq

k¸2 i0

k¸1 j2

fkj 1p1ⁱbmj b1^kⁱ^jq

k¸1 s1

m₂pf_sbf_k_sq .

For the second summand we use (4.23) and Df₁ 0, that is r

m₂ pf_kb pDhf₁qq rm₂ pf_kb ppid Tqf₁qq rm₂ pf_kbf₁q by (4.24). For (4.27) we deduce

p1q^|^x¹^{| |}^x^k^| ^k¹pDf_k ₁qpx₁, ..., x_k ₁q

k¸1 s1

m2 p rm2pfsbfksq bhf1q p1q^|^x¹^{| |}^x^k^| ^kmr2 pfkbf1q

px1, ..., xk 1q . Using that the de Rham loop product and thus rm₂ is associative implies

p1q^|^x¹^{| |}^x^k^| ^k¹pDf_k ₁qpx1, ..., x_k ₁q

k¸1 s1

p1q^|^x¹^{| |}^x^s^| ^smr2 pfsb rm2pfksbhf1qq p1q^|^x¹^{| |}^x^k^| ^kmr2 pfkbf1q

px1, ..., xk 1q

By denition (4.26) we have r

m2 pf_ksbhf1q p1q^|^x¹¹^| ^... ^|^x¹^k^s^| ^k^s¹f_ks ₁

for x¹_i x_{s i}, that is we get

p1q^|^x¹^{| |}^x^k^| ^k¹pDf_k ₁qpx₁, ..., x_k ₁q

k¸1 s1

p1q^|^x¹^{| |}^x^k^| ^k¹mr2 pfsbfks 1q p1q^|^x¹^{| |}^x^k^| ^k¹mr2 pfkbf1q

px1, ..., xk 1q p1q^|^x¹^{| |}^x^k^| ^k¹

k¸1 s1

m₂ pf_sbf_k_s ₁q mr₂ pf_kbf₁q

px₁, ..., x_k ₁q p1q^|x¹^{| |x}^k^| ^k1

¸k s1

m₂ pf_sbf_k_s ₁q

px₁, ..., x_k ₁q

which is

p1q^|^x¹^{| |}^x^k^| ^k¹U_k ₁px₁, ..., x_k ₁q since mj|_p^À

aPA

HpL^aXqq^b^k 0 for1¤j ¤kby the induction hypothesis.

Proof of theorem 4.16 : We use the notation from section 3.3.

For the rst operations we set

λ₁ 0V₁ and φ₁ ι ,

where ι:HpL^aXq Ñ pC¹q^a for a PA and ι: HpL⁰Xq Ñ pC¹q⁰ are cycle choosing homomorphisms. Thus equation (3.11) is satised, namely

φ₁λ₁V₁0Dι rλ₁φ₁ .

The operationλ2 rrλ2i^b²s on HpLXq is the loop bracket up to sign due to how λr2 : de Rham loop bracket

are constructed by Irie.

The recursive construction of Kadeishvili yields

V2px, yq rλ2pφ1pxq, φ1pyqq

for x, yPHpLXq. Therefore by construction forabPA and xPHpL^aXq we get

V₂px, yq

0 , for yPHpL^bXq

P pC¹q^a , for yPHpL^aXq ory PHpL⁰Xq

since pC¹q^a and HpL^aXq contains families/classes of loops that are positive iterates of the primitive nontrivial homotopy classa.

Forλ2: rV2s we get

λ₂px, yq

0 , for yPHpL^bXq

PHpL^aXq , for yPHpL^aXq or yPHpL⁰Xq which allows to dene φ2 such that

φ₂px, yq

0 , fory PHpL^bXq

P pC¹q^a , fory PHpL^aXq or yPHpL⁰Xq Analogously we get

λ2px, yq PHpL⁰Xq and φ2px, yq P pC¹q⁰ for x, yPHpL⁰Xq.

We end up with (3.11), namely

φ₁λ₂V₂ rλ₁φ₂ .

It remains to prove piiiq. We perform the inductive step forkÑk 1.

Assume all operations and morphisms are constructed up to degreek and that

λ_lpc1, ..., c_lq

0 , pIq

PHpL^aXq , pIIq

PHpL⁰Xq , pIIIq (4.28) and

φlpc1, ..., clq

0 , pIq P pC¹q^a , pIIq

P pC¹q⁰ , pIIIq (4.29) for all 1¤l¤k. Here condition pIq means

pIq p Di, jP t1, ..., lu such that ci PHpL^aXq and cj PHpL^bXq for abPA , pIIq means

pIIq p @iP t1, ..., lu we either have c_iPHpL⁰Xq orc_i PHpL^aXq foraPA and there exists at least onei0P t1, ..., lu such that ci0 PHpL^aXq and pIIIq means

pIIIq p @iP t1, ..., lu we havec_i PHpL⁰Xq .

We prove that (4.28) and (4.29) hold forλk 1 andφk 1 which then proves piiiq of the theorem, namely that

λlpc1, ..., clq 0 if Di, j such thatci PHpL^aXq andcj PHpL^bXq forabPA and for alll¥1.

The recursive construction of Kadeishvili yields

V_k ₁pc1, ..., c_k ₁q ¸

σ, p qk 1

cσrλ2pφppc_σ_p₁_q, ..., c_σ_p_p_qq, φqpc_σ_p_p ₁_q, ..., c_σ_p_k ₁_qqq

τ, 1 l k 1

cτφkl 2pλlpcτp1q, ..., c_τ_p_l_qq, cτpl 1q, ..., c_τ_p_k ₁_qq .

The multiplicities, the signs and in particular the question which σ and τ are used, shues or permutations, is an important issue in general. We may bypass these ques-tions since the statements above will hold independently for each summand.

Since only morphisms and operations of degree ¤kare involved we apply the induction hypothesis and get

Vk 1pc1, ..., ck 1q

0 , pIq P pC¹q^a , pIIq P pC¹q⁰ , pIIIq Sinceλ_k ₁: rV_k ₁s we get

λ_k ₁pc₁, ..., c_k ₁q

0 , pIq PHpL^aXq , pIIq PHpL⁰Xq , pIIIq

According to the denition of Kadeishvili φ_k ₁ is dened such that

Dφ_k ₁pc₁, ..., c_k ₁q:V_k ₁pc₁, ..., c_k ₁q φ₁pλ_k ₁pc₁, ..., c_k ₁qq . Sinceφ₁ satises (4.29) we can chooseφ_k ₁ such that

φ_k ₁pc₁, ..., c_k ₁q

0 , pIq P pC¹q^a , pIIq P pC¹q⁰ , pIIIq

This nishes the inductive step and proves piiiq namely that for pIq we have λ_k ₁pc₁, ..., c_k ₁q 0 .

Im Dokument Contributions to the string topology of product manifolds (Seite 88-99)