The homotopy transfer construction for Lie algebras

Notice the considerations about algebras and A₈-algebras presented in section 3.1.

Here we only recall ideas of Appendix A3. of [14], chapter 10. of [27] and section 4 of [25]. The interested reader is referred to these sources for more details.

AnL₈-algebra overRconsists of a graded real vector spaceC À

mPZ

C^mand operations λ_n: ΛⁿCÑC pn ¥1q

of degree |λ_n| n2(homological convention) such that

¸

n1 n2n 1

p1qⁿ² ¸

ρPSn

ρp1q ... ρpn1q ρpn1 1q ... ρpnq

λ_n2pλ_n1pa_ρp1q, ..., a_ρpn1qq, a_{ρpn1 1q}, ..., a_ρpnqq 0 (3.8)

where 1is determined by a₁^...^a_n a_ρp1q^...^a_ρpnq.

HereΛⁿCTⁿC{pabb p1q^|a||b|bbaq denotes the nth exterior power of C and S_n is the symmetric group.

As forA₈-algebras there is an equivalent approach in terms of a bar construction. The concept is equivalent to

pSpCr1sq, η, lq

being a dierential coalgebra structure, that is pl pl 0. Precisely speaking on SpCr1sq:à

k¥1

S^kpCr1sq:à

k¥1

pCr1s b bCr1sq{

whereSⁿCTⁿC{pabb p1q^|a||b|bbaq, we dene

l_kpc₁ c_rq ¸

ρPSr

1

k!prk!qpσ₁λ_kσ¹_k qpc_ρp1q c_ρpkqq bc_{ρpk 1q}b bc_ρprq if r¥k and zero else. Here we use the isomorphism

pΛ^kCqrksÝÑ^σk S^kpCr1sq

a₁^...^a_k ÞÝÑ p1q^°pkiq|ai|a₁ a_k where we used degrees inC for |a_i| for1¤i¤k.

AnL₈-algebra morphism between L₈-algebras pC,tλ_kuk¥1q and pC¹,tλ¹_kuk¥1q is a se-quence of maps tφ_k: Λ^kC ÑCuk¥1 of degree |φ_k| k1 that satisfy

e^gl l¹e^g (3.9)

in the bar construction, where

g_k :σ₁φ_kσ_k¹ :S^kpCr1sq Ñ C¹r1s and

e^g :SpCr1sq ÑSpC¹r1sq; c₁c_kÞÑ ¸

k1 ...krk

¸

ρ

1

r!k₁!...k_r!pg_k1b...bg_krqpc_ρp1qb...bc_ρpkqq. Similar to the fact that dierential graded algebras can be viewed asA₈-algebras, it is possible to interpret a dierential graded Lie algebra pC, d,t,uq as an L₈-algebra.

In fact we have

ta, bu p1q^|a||b|tb, au and when setting

λ₁ :d, λ₂pa, bq: p1q^|a|ta, bu and λ_k¥3 :0,

for n1,2,3 equation (3.8) translates into dd0,

dta, bu tda, bu p1q^|a|ta, dbu,

ta,tb, cuu tta, bu, cu p1q^|a||b|tb,ta, cuu .

For general L₈-algebras the Jacobi identity just holds up to homotopy given by λ₃. So when passing down to homology via the boundaryλ₁ Jacobi identity strictly holds.

As for A₈-algebras we can transfer L₈-algebra structures from one complex C to a quasi-isomorphic complex B and thus in particular to homology HpCq. Generally speaking we want to transfer structure from C to a homotopy retractB.

Theorem 3.10 ([27] Theorem 10.3.2., Theorem 10.3.8., Theorem 10.3.9.) Let pB, d_Bq and pC, d_Cq be chain complexes such that

pB, d_Bqoo _p^{i //}pC, d_Cq ý^h

pi idB ppsurjective and i injectiveq ip idC d_Ch hd_C piis chain homotopy equivalenceq

Suppose trλ_nun¥1is anL₈-algebra structure onCwith rλ₁ d_C. ThenBis equipped with an induced L₈-algebra structure tλ_nun¥1 with λ₁ d_B pictorially given by gure 3.3, and φ₁ :i extends to a morphism tφ_nun¥1 of L₈-algebras.

Figure 3.3: Transferring L₈-algebras

Inspired by the work of Kadeishvili in [21] there is a recursive construction for λ_n without using the stated trees and the global homotopyhdisplayed on the inner edges above. In the following we prefer that approach since we do not want to specify the homotopy h.

As described in [8] the homotopy transfer may be done recursively without specifying the homotopy h in the case that we transfer to homology, namely for the set-up

pHpC, d_Cq, d0q ^{i //}pC, d_Cq .

oo p

This is done in much more generality in Theorem 6.1. of [8]. Restricting to the case of L₈-algebras, it is proven that a given L₈-algebra pC,trl_kuk¥1q and in particular a Lie algebra structure (that is rl_k 0 for k ¥ 3) transfers to an L₈-algebra structure pHpCq,tlkuk¥1q and further that there exists an L₈-algebra morphism

g :HpCq ÑC

such thatg₁ is a quasi-isomorphism. The morphismg is called a 8-quasi-isomorphism.

Analogously to equation (3.3) for the homotopy transfer forL₈-algebras Lemma 2.9.

of [8] yields a relation between theL₈-algebra operations and the morphism g, namely

g_kl₁ rl₁g_k g₁l_k 1

k!rl_kg^d₁^k R_kpg, l,rlq 0 (3.10) where

g_k1 d dg_kipc₁ c_lq : ¸

σPSk

pσq

k₁! k_i!g_k1pc_σp1q c_σpk1qq g_kipc_{σpk1 ... ki1 1q} c_{σpk1 ... kiq}q and pσq is determined by c_σp1q c_σpkqpσqc₁ c_k. Here the morphisms R_kpg, l,rlq only contain components l_k1, rl_k1, g_k1 with k¹   k. In particular R₁ 0 and R₂ 0 since rl₁g₁ 0l₁.

Sincel₁ 0 equation (3.10) simplies to rl₁g_kg₁l_k 1

k!rl_kg₁^dk R_kpg, l,rlq

When we are in the special case that C is just a Lie algebra we have rl_k 0for k ¥3 and thus may write

rl₁g_k d_cg_kg₁l_k

$'

&

'%

rl₁g₁ d_ci0 , k1

1

2rl₂g₁^d2 , k2 R_kpg, l,rlq , k¥3

:g₁ l_k V_k

and analogously

rλ₁ φ_kd_cφ_k φ₁ λ_k σloooooomoooooon₁ V_kσ_k¹

:Vk

. (3.11)

Chapter 4

Higher string topology via homotopy transfer

Throughout this chapter we work with real coecients and loop spaces consisting of smooth loops. When talking about string topology on chain level we have to specify which chain complex we are working with such that its homology yields the singular homology ofLM. Further in full strictness string topology operations are only dened on the level of homology via homotopy theoretical considerations as in [11]. The def-inition of [5] namely by dening them geometrically on chain level and then let them descend to homology still lacks the specication which chain model one should use.

For performing the homotopy transfer construction later we have to think of how the initially only partially dened operations can be fully dened such that the chain com-plex becomes a dierential graded algebra respectively dierential graded Lie algebra.

By using the work of Irie (cf. [20]) we get a chain level version of the loop product and the loop bracket. We then let these structures descend to homology HpLMq which yields an A₈{L₈-algebra structure on HpLMq for general closed and oriented manifoldsM.

Remark that we rely on version v2 of Irie's work [20] in the following. The most recent version of this document is version v4 with the title "`A chain level Batalin-Vilkovisky structure in string topology via de Rham chains"'.

A dierent and more algebraic approach would be to work in the language of operads.

We are not discussing these methods here but refer to [24] and [36]. There it is proven that there exists a functor converting partial algebras into algebras such that both are quasi-isomorphic as partial algebras. Especially Theorem 2.7.3. of [36] states that the complex of chains of the free loop space can be equipped with a Lie algebra structure induced by the loop bracket. We do not pursue this approach since we are interested in actually computing the operations on chain level. This would be harder when working on that more algebraic level.

Finally when we understand the homotopy algebra structures, in particular the L₈ -algebra, we are able to use Fukaya's theorem 1.1 in a meaningful way and prove that a product of a hyperbolic manifold and a simply connected manifold does not embed as a Lagrangian submanifold intoCⁿ.

69

4.1 De Rham homology of LM

According the work of Irie in [20] for a given manifold M it is possible to dene a chain complex for LM C⁸pS¹, Mq which becomes a dierential graded algebra and a dierential graded Lie algebra (with a degree 1operator∆) which further descends to the known BV-algebra structure on homology dened in [5]. We briey recall the author's ideas. This is done in order to adapt ideas and then discuss the case for LpNsc. M_K 0q in the next section where N is simply connected of dimensionn ¥0 and M has negative sectional curvature and is of dimension m ¥ 3. As usually in string topology N and M are assumed to be closed and oriented.

In the following we refer to denitions and results of [20].

A dierentiable space is a set X equipped with a dierentiable structure PpXq: tpU, φq | U PU, φ:U ÑX is a plotu , where U : —

0¤n¥1k¤n

U_n,k and U_n,k is the set of k-dimensional oriented C⁸-submanifolds of Rⁿ without boundary. The collection of plots tφ : U ÑXu is required to have the following properties:

(i) If θ :U¹ ÑU aC⁸-submersion for U¹ P U and pU, φq P PpXq, then pU¹, φθq P PpXq

(ii) If φ : U Ñ X is a map with U P U such that there is an open covering pU_αqαPI

of U such that pUα, φ|Uαq PPpXq for all αPI, then pU, φq PPpXq.

A manifold M is a dierentiable space by specifying φ : U Ñ M to be a plot if φ is smooth, that is

pU, φq PPpMq:ôφ PC⁸pU, Mq

A subset X₁ Ñ^ι X₂ of a dierentiable space X₂ is a dierentiable spaces by specifying a map φ:U ÑX₁ to be a plot if pU, ιφq PPpX₂q.

A map between dierentiable spaces f :X ÑY is smooth if pU, φq PPpXq implies pU, f φq PPpYq .

By denition the inclusion X1

Ñι X2 is a smooth map.

Two such maps f, g are smoothly homotopic if a smooth map h : X R Ñ Y exists such that

hpx, sq

"

fpxq , s 0 gpxq , s¡1 .

Remark that we have a canonical dierentiable structure on products of dierentiable spaces. A map is a plot if all its projections are plots of the particular factors.

In the following we want to treat free loop spaces. ForM a smooth closed and oriented manifold andLM C⁸pS¹, Mq we dene a dierentiable structure as follows:

pU, φq PPpLMq:ôevφP C⁸pU S¹, Mq where pevφqpu, tq:φpuqptq. Remark that by denition evaluation mapsLM ^evÑ^t M;γ ÞÑγptq are thus smooth.

Further the energy functional

E :LM ÑR (4.1)

γ ÞÑ

»

S¹

| 9γ|² is smooth for the dierentiable structures dened above.

For a dierentiable space pX,PpXqq we dene the de Rham chain complex C_k^dRpXq:RxZ_kpXqy{Z_kpXq pk ¥0q

where the vector spaceRxZ_kpXqy is generated by the set

Z_kpXq: tpU, φ, ωq | pU, φq PPpXq, ω PΩ^dim_c ^UkpUqu, whereΩⁱ_cpUq is the vector space of compactly supported i-forms onU. We mod out the subspaceZkpXq generated by vectors:

• apU, φ, ωq pU, φ, aωq for aPR

• pU, φ, ωq pU, φ, ω¹q pU, φ, ω ω¹q

• pU, φ, π_!ωq pU¹, φπ, ωq, where π_! : Ω^r_cpU¹q ÑΩ^r_c^{dimU1 dim U}pUq is the integra-tion along the ber dened for C⁸-submersions π:U¹ ÑU

The linear degree 1map

B rpU, φ, ωqs: rpU, φ, dωqs

denes a boundary. We dene de Rham homology as the homology H^dRpXq:HpC^dRpXq,Bq .

An augmentation is given by rpU, φ, ωqs ÞÑ ³

U

ω for rpU, φ, ωqs PC₀^dRpXq Smooth mapsf :X ÑY between dierentiable spaces induce chain maps

fprpU, φ, ωqsq: rpU, f φ, ωqs .

The de Rham chain complex is indeed functorial here since smoothly homotopic maps induce chain homotopic maps as shown in Proposition 2.5. of [20].

Next we want to compare Irie's construction with standard singular homology.

A map ρ : ∆^k Ñ X is strongly smooth if either k 0 or if k ¡ 0 and there exists a neighbourhood U of

∆^k tpt₁, ..., t_kq PR^k | 0¤t₁ ¤...¤t_k ¤1u €R^k

and a smooth map ρ : U Ñ X such that ρ|∆^k ρ. For a dierentiable space we can dene the chain complex of strongly smooth maps

S^smpXq € SpXq à

k¥0

R x Mapp∆^k, Xq y

as the sub-complex generated by strongly smooth maps inside the singular chain com-plex.

Lemma 4.1 (e.g. Theorem 18.7 of [26])

For a smooth nite dimensional manifold X the inclusion S^smpXq ãÑ SpXq

is a quasi-isomorphism. It yields an isomorphism

H^smpXq HpXq . (4.2)

Remark that∆^kcarries the canonical structure of a dierentiable space as it is a subset of R^k.

Lemma 4.2 (Lemma 2.6. and Proposition 3.2. of [20]) There exist u_kPC_k^dRp∆^kq for allk PN0 such that the map

S_k^smpXq Ñ C_k^dRpXq σÞÑσpu_kq

for X a smooth nite dimensional manifold is a chain map and yields an isomor-phism

H^smpXq H^dRpXq (4.3)

that is not depending on the choice of pu_kqk¥0.

When combining both Lemmas we conclude that de Rham homology computes real singular homology for nite dimensional smooth manifolds.

Proposition 4.3

For X a smooth nite dimensional manifold there exists an isomorphism

H^dRpXq HpXq . (4.4)

We want a similar result for free loop spaces of nited-dimensional smooth Riemannian manifolds M that are closed and oriented. That is we want an isomorphism

H^dRpLMq HpLMq.

By choosing a strictly increasing sequence pE_jqj¥1 such that lim

jÑ8E_j 8 we dene the energy ltration of LM via

LM^Ej : tγ PLM | Epγq   Eju

where we used the energy as dened in (4.1). Inclusion of subspaces LM^Ei ãÑLM^Ej (j ¡i) provides a directed system which in turn yields homomorphisms

limÝÑ

jÑ8

H_kpLM^Ejq ÑH_kpLMq limÝÑ

jÑ8

H_k^smpLM^Ejq ÑH_k^smpLMq (4.5) limÝÑ

jÑ8

H_k^dRpLM^Ejq ÑH_k^dRpLMq .

Remark that pEpxq E_jqj¥1 is a sequence of decreasing smooth functions, that is pEpxq E₁q ¥ pEpxq E₂q ¥... for all xPLM ,

and lim

jÑ8pEpxq Ejq 8 for all x P LM. Therefore results of chapter 2.7. of [20]

can be applied.

Lemma 4.4 (chapter 3.3. of [18]; Lemma 2.8. and Lemma 2.10. of [20]) For the loop space LM of a nite dimensional, closed and oriented Riemannian manifoldsM with the energy ltration

LM^Ej : tγ PLM |Epγq   E_ju the inclusion induces isomorphisms

limÝÑ

jÑ8

H_kpLM^Ejq Ñ H_kpLMq limÝÑ

jÑ8

H_k^smpLM^Ejq Ñ H_k^smpLMq (4.6) limÝÑ

jÑ8

H_k^dRpLM^Ejq Ñ H_k^dRpLMq .

Proof (sketch) : Represent a cycle in LM by singular simplices. The union of their images is a compact set in LM where the energy functional E attains a maximum Ej0 and thus the cycle is a cycle in LM^Ej0. This proves surjectivity.

Injectivity follows similarly since a bounding chain in LM of a cycle in LM^Ej1 is compact and thus lies in someLM^Ej2 for j₁¤j₂.

In order to prove that de Rham homology computes singular homology for free loop spaces of nite dimensional smooth manifolds it is therefore enough to show that

limÝÑ

jÑ8

H_kpLM^Ejq Ð_jlimÝÑ_Ñ8H_k^smpLM^Ejq Ñ_jlimÝÑ_Ñ8H_k^dRpLM^Ejq (4.7) are isomorphisms.

In [20] this is done by approximating the free loop space LM by nite dimensional smooth manifoldsF_N^EM. By previous considerations (4.2) and (4.3) we know that we have isomorphisms

limÝÑ

jÑ8

H_kpF_N^Ej

jMq Ð _jlimÝÑ_Ñ8H_k^smpF_N^Ej

jMq Ñ_jlimÝÑ_Ñ8H_k^dRpF_N^Ej

jMq. (4.8) So it remains to clarify how the approximations F_N^EM are dened and then to show that (4.7) is equivalent to (4.8).

Finite dimensional approximations of LM

Remark that M is equipped with a Riemannian metric, so that we can measure dis-tances. We approximate a loop by a nite number of points on it, that is we dene

F_NM : tpx₀, ..., x_Nq PM^N | x₀ x_Nu,

F_N^E0M : tx px₀, ..., x_Nq PF_NM | Epxq:N ¸

0¤j¤N1

dpx_j, x_{j 1}q²  E₀u . The approximations carry the canonical dierentiable structure as subsets ofM^N. Lemma 4.5 (Lemma 4.3. of [20])

For a sequence E_j Ñ 8 of strictly increasing positive real numbers there exists a sequence N_j Ñ 8 of integers such that the evaluation map

e_N :LM ÑLM_N (4.9)

γ ÞÑ pγp0q, γp1{Nq, γp2{Nq, ..., γp1qq induces an isomorphism

limÝÑ

iÑ8

H^#pLM^Eiq

limÝ

iÑÑ8

H^#pe_Niq

// limÝÑ_iÑ8H^# pF^Ei_N

iMq . (4.10)

Here # either means 'de Rham homology' or 'smooth singular homology' or 'sin-gular homology'.

The Lemma combined with the isomorphisms in (4.8) imply that limÝÑ

jÑ8

H_kpLM^Ejq Ð_jlimÝÑ_Ñ8H_k^smpLM^Ejq Ñ_jlimÝÑ_Ñ8H_k^dRpLM^Ejq

are isomorphisms. Since we already proved thatLM^Ei ãÑ LM induces isomorphisms on homology for singular homology, smooth singular homology and de Rham homology we conclude that

H_kpLMq ÐH_k^smpLMq ÑH_k^dRpLMq

are isomorphisms that is de Rham homology computes singular homology for free loop spaces of nite dimensional smooth Riemannian manifoldsM that are closed and ori-ented.

Corollary 4.6

ForM a smooth nite dimensional manifold there exists an isomorphism

H^dRpLMq HpLMq. (4.11) Proof of Lemma 4.5 : The evaluation map

e_N :LM ÑLM_N (4.12)

γ ÞÑ pγp0q, γp1{Nq, γp2{Nq, ..., γp1qq

is smooth by denition and eNpLM^E0q € F_N^E0M by using the Cauchy-Schwarz in-equality, namely forγ PLM^E0 one has

Epe_Npγqq N ¸

0¤j¤N1

dpx_j, x_{j 1}q²N ¸

0¤j¤N1

i 1

»N i N

| 9γ|

2

¤N ¸

0¤j¤N1

i 1

»N i N

1²

i 1

»N i N

| 9γ|²

¸

0¤j¤N1

i 1

»N i N

| 9γ|²

»

S¹

| 9γ|²  E0 .

ForE₀ xed we choose N₀ suciently large such that aE₀{N₀  r_M

whererM is the injectivity radius that is positive since M is closed.

Then

dpxj, xj 1q  d ¸

0¤j¤N1

dpxj, xj 1q² a

E0{N0 rM ,

so that there is a geodesic connectingxj andxj 1 which we denote byγxj,xj 1. These geodesics will soon be further subdivided into m parts. We x m ¡ 0. For given energies0 E₀ E¹₀ we choose δ¡0 such that

p1 δq⁴  E₀¹{E₀ . Our next goal is to dene a map

g0 :F_N^E0

0M ÑLM^E01 (4.13)

that is smooth (in the sense above) and continuous (in the sense of Whitney C⁸ -topology). For that we need a map µ:r0,1s Ñ r0,1s that satises

(i) 0¤µ¹ptq ¤1 δ (ii) µpi{mq i{m

(iii) µis constant near 0 and 1 . Now we set g0px0, ..., xN0q γ where

γptq

$' '&

''

%

γx0,x1pµpN0t0qq ; tP r0,1{N0s γ_x1,x2pµpN₀t1qq ; tP r1{N₀,2{N₀s

γxN01,xN0pµpN0t pN01qqq ; tP rN01{N0,1s .

Notice that property piq ofµ impliesEpγq ¤ p1 δq²Epxq p1 δq²E0 E₀¹. We dene

im:F_N^E0

0M ÝÑ^g0 LM^{E01 e}ÝÑ^mN0F_mN^E0

0M

px0, ..., xN0q ÞÑγ ÞÑ pxloooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooon0, γx0,x1p1{mq, ..., γx0,x1p1q γx1,x2p0q x2, ..., ..., xN0q

mN0 1

.

One further checks that

LM^{E0 incl. //}

eN0

LM^E01

F_N^E0

0M

g0

::

commutes up to homotopy. This is done in chapter 4 of [20]. Roughly speaking since N₀ is suciently large points of incl.(γ) and pg₀e^E_N⁰

0qpγq in LM^E10 can be connected by geodesics. This denes a smooth homotopyγsconnecting these two loops. Further we have Epγ_sq ¤ p1 δq⁴E₀  E₀¹.

We end up with smooth (in the sense above) and continuous (in the sense of Whitney C⁸-topology) maps tting in the diagram

LM^{E0 incl. //}

eN0

LM^E10 :LM^E1

emN0:eN1

F_N^E₀⁰M

g0

99

im

//F^E10_mN0M :F^E1_N1M

that commutes up to homotopy.

Continuing the construction inductively we get a sequence ^//LM^{Ej //}

LM^{Ej 1}

//

^//F_N^Ej

jM

gj

:://F^{Ej 1}_N

j 1M

gj 1

;;// .

In total we get an isomorphism

limÝÑ

iÑ8

H^#pLM^Eiq

limÝÑ

iÑ8

H^#pe_Niq

// limÝÑ

iÑ8

H^#pF_N^Ei

iMq

limÝÑ

iÑ8

H^#pgiq

oo .

Im Dokument Contributions to the string topology of product manifolds (Seite 68-81)