Table Of ContentA CHEN MODEL FOR MAPPING SPACES
AND THE SURFACE PRODUCT
9
0
0 GRE´GORYGINOT,THOMASTRADLER,ANDMAHMOUDZEINALIAN
2
y
a
M
Abstract. We develop a machinery of Chen iterated integrals for higher
Hochschildcomplexeswhicharecomplexeswhosedifferentialsaremodeledby
4
anarbitrarysimplicialsetmuchinthesamewaythattheordinaryHochschild
1
differentialismodeledbythecircle. Weusethesetogivealgebraicmodelsfor
generalmappingspacesanddefineandstudythesurfaceproductoperationon
]
A thehomologyofmappingspacesofsurfacesofallgeneraintoamanifold,which
is an analogue of the loop product in string topology. As an application we
Q
showthatthisproductishomotopy invariant. WeproveHochschild-Kostant-
. Rosenbergtypetheoremsandusethemtogiveexplicitformulaeforthesurface
h
productofoddspheresandLiegroups.
t
a
m
Contents
[
1. Introduction 1
1
v 2. Chen’s iterated integrals for Hochschild complexes over simplicial sets 6
1 2.1. Higher Hochschild chain complex over simplicial sets 6
3 2.2. Definition of Chen’s iterated integrals 8
2
2.3. Examples: the circle, the sphere and the torus 11
2
2.4. The cup product for the mapping space Map(Y,M) 15
.
5 2.5. Chen’s iterated integrals as a quasi-isomorphism 19
0
3. String topology product for surfaces mapping spaces 21
9
3.1. A Hochschild model for the surface of genus g 21
0
: 3.2. The “string topology” product for surfaces 23
v
3.3. Surface Hochschild cup product 29
i
X 3.4. Topological identification of the cup product 42
r 3.5. Proof of Theorem 3.4.2 43
a
4. Surface Hochschild (co)homology of symmetric algebras 51
4.1. Reduction to Hochschild complexes over a square and a wedge of
circles 51
4.2. HKR type Theorem for wedges of circles 52
4.3. HKR type Theorem for surfaces 57
4.4. The surface product for Lie groups 62
References 68
1. Introduction
An element of the Hochschild chain complex CH (A,A) of an associative alge-
•
bra A is, by definition, givenby elements in a multiple tensor product A⊗···⊗A,
1
2 G.GINOT,T.TRADLER,ANDM.ZEINALIAN
whose tensor factors all consist of the algebra A. When defining the differential
D :CH (A,A) →CH (A,A), it is however instructive to picture this linear se-
• •−1
quenceoftensorproductsasplacedinacircularconfiguration,sincethedifferential
multiplies any two adjacent tensor factors, with its last term multiplying the last
factor of the sequence to the first factor,
⊗ ai ⊗ ⊗ai+1·ai⊗
D
±
⊗ ⊗
⊗an a1 mfuolrtiaplllyPia=i0t,.o..a,ni+,1 ⊗an a1
⊗a ⊗ ⊗a ⊗
0 withan+1≡a0 0
As it turns out this is not just a mnemonic device but rather an explanation of
the fundamental connection between the Hochschild chain complex and the circle,
which, for instance gives rise to the cyclic structure of the Hochschild chain com-
plex and thus to cyclic homology, see [L]. This connection is also at the heart of
the relationship between the Hochschild complex of the differential forms Ω•M on
a manifold M, and the differential forms Ω•(LM) on the free loop space LM of
M, which is the space of smooth maps from the circle S1 to the manifold M; see
[GJP]. Atthe coreofthis connectionisthe factthattheHochschildcomplexisthe
underlying complex of a simplicial module whose simplicial structure is modelled
on a particular simplicial model S1 of the circle. The principle behind this can be
•
fruitfully used to construct new complexes whose module structure and differen-
tial are combinatorially governed by a given simplicial set X , much in the same
•
way that the ordinary Hochschild complex is governed by S1; see [P]. However
•
carryingthe constructionto higher dimensional simplicialsets turns out to require
(associative and) commutative algebras. The result of these constructions define
for any (differential graded) commutative algebra A, any A-module N, and any
pointed simplicial set X•, the (higher) Hochschild chain complex CH•X•(A,N) of
A and N over X aswellastheHochschildcochaincomplexCH• (A,N)overX ;
• X• •
see[P,G]. TheseHochschild(co)chaincomplexesarefunctorialinallthreeoftheir
variables A, N and X .
•
The analogy with the usual Hochschild complex and its connection to the free
loop space is in fact complete, since the Hochschild complex CHX•(Ω•M,Ω•M)
•
overasimplicialsetX providesanalgebraicmodelofthe differentialformsonthe
•
mapping space MX = {f :X →M}, where X = |X | is the geometric realisation
•
ofX . ThisisoneofthemainresultofSection2ofthispaper;seeSection2.5. The
•
main tool to prove this result is a machinery of iterated integrals that we develop,
which in turn enables us to work with the associative and commutative algebra
of differential forms yielding a quasi-isomorphicm ItX• : CHX(Ω•M,Ω•M) →
•
Ω•(MX), for any k-dimensional simplicial set X and k-connected manifold M;
•
see Proposition 2.5.3, and Corollary 2.5.5 for the dual statement. Further, for any
simplicial set X and (differential graded) commutative algebra A, the Hochschild
•
chaincomplexCHX•(A,A)hasanaturalstructuredifferentialgradedcommutative
•
algebra given by the shuffle product sh (Proposition 2.4.2). We show that the
X•
iterated integral ItX• : (CHX(Ω•M,Ω•M),sh → (Ω•(MX),∧) is an algebra
• X•)
A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT 3
map sending the shuffle product to the wedge product of differential forms on the
mapping space; see Proposition 2.4.6.
Two of the most important features of Hochschild (co)chain complexes over
simplicial sets is their naturality in the simplicial set X , and furthermore that
•
two simplicial models of quasi-isomorphic spaces have naturally quasi-isomorphic
Hochschild (co)chain complexes, see [P]. In particular one (usually) obtains many
different models to study CH•X•(A,N) for a given space X = |X•|. These facts
can be combined to carry geometric or topologicalconstructions to the Hochschild
complex, which is applied in Sections 3 and 4 to study Hochschild (co)homology
over compact surfaces Σg of genus g.
The collection of compact surfaces of any genus is naturally equipped with a
product similar to the loop productof string topology[CS], also see [S2]. The idea
behind this product, that we call the surface product, is shown in the following
picture.
(1.1)
wedge pinch
InSection3.1, wedescribe anexplicitsimplicialmodelfor the stringtopologytype
operation induced by the map
Map(Σg,M)×Map(Σh,M)←ρi−n Map(Σg∨Σh,M)−ρo→ut Map(Σg+h,M)
coming from the above picture (1.1). More precisely, we obtain a surface product
⊎:H (Map(Σg,M))⊗H (Map(Σh,M))→H (Map(Σg+h,M)), which is
• • •+dim(M)
given by the composition of the Umkehr map (ρ ) and the map induced by ρ ,
in ! out
⊎:H (Map(Σg,M))⊗H (Map(Σh,M))
• •
(−ρi→n)! H (Map(Σg∨Σh,M))(ρ−ou→t)∗ H (Map(Σg+h,M)).
• •
We prove that the surface product makes
H (Map(Σg,M)),⊎ = H (Map(Σg,M)),⊎)
• •+dim(M)
g g
(cid:0)M (cid:1) (cid:0)M
into an associativebigradedalgebrawith H (Map(Σ0,M)) in its center; see Theo-
•
rem3.2.2andProposition3.2.5. Therestrictionofthesurfaceproducttogenuszero
(i.e. spheres),H (Map(Σ0,M)),coincideswiththeBranetopologyproductdefined
•
by Sullivan and Voronov H (Map(S2,M))⊗2 → H (Map(S2,M)) see [H, CV].
• •
In fact, in these papers it is shown, that H (MSn) is an algebra over
•+dim(M)
4 G.GINOT,T.TRADLER,ANDM.ZEINALIAN
H (fD ), the homology of the framed n-disc operad; see also [S2] for related
• n+1
algebraic structures.
InSection3.3weapplythe machineryof(higher)Hochschildcochaincomplexes
over simplicial sets to give a fully algebraic description of the surface product. In
fact, for positive genera, we define an associative cup product ∪ : CH• (A,B)⊗
Σg•
CH• (A,B)→CH• (A,B) for the Hochschild cochains over surfaces, where B
Σh• Σ•g+h
is a differential graded and unital commutative A-algebra, viewed as a symmetric
bimodule;seeDefinition3.3.2. Theconstructionofthecup-productisbasedonthe
·
fact that for any pointed simplicial sets X and Y , the multiplication B⊗B →B
• •
inducesacochainmapCH• (A,B)⊗CH• (A,B)→∨ CH• (A,B)whichcan
Σg• Σh• (Σg∨Σh)•
Pinch∗
be composed with the pullback CH• (A,B) →g,h CH• (A,B) induced
(Σg∨Σh)• Σ•i+j
byasimplicialmodelPinch :Σg+h →Σg∨Σh forthemappinchinFigure(1.1).
g,h • • •
However, this product is initially only defined for surfaces of positive genera,
and more work is required to include the genus zero case of the 2-sphere in this
framework. To this end, we first recall a cup product for genus zero defined in
[G], and then define a left and right action, ∪˜, of CHΣ0•(A,B) on CHΣg•(A,B).
• •
Taking advantage of the fact that one can choose different simplicial models for
a given space, we then show that ∪˜ is equivalent after passing to homology to an
operationsimilar to the cup-product but defined onsome other different simplicial
models for the sphere and surfaces of genus g; see Definition 3.3.13 and Propo-
sition 3.3.14. Putting everything together, we obtain a well-defined cup product
∪ : HH• (A,A) ⊗2 → HH• (A,A) for all genera on cohomology;
g≥0 Σg• g≥0 Σg•
seeTheorem3.3.18. Moreprecisely,weprovethatforadifferentialgradedcommu-
(cid:0)L (cid:1) L
tative algebra (A,d ),
A
i) thecupproductmakes HH• (A,A)intoanassociativealgebrawhich
g≥0 Σg•
is bigraded with respect to the total degree grading and the genus of the
L
surfaces and has a unit induced by the unit 1 of A.
A
ii) HH• (A,A) lies in the center of HH• (A,B).
Σ0• g≥0 Σg•
ThecupproductisfunctorialwithrespectLtobothargumentsAandB (seePropo-
sition 3.3.20).
The connectionto topologyisprecise,that is the cup-productis amodel forthe
surface product. We prove in Theorem 3.4.2 that for a 2-connected compact man-
ifold M, the (dualized) iterated integral ItΣ• : ( H (Map(Σg,M)),⊎) →
g≥0 −•
( HH−•(Ω,Ω),∪) is an isomorphism of algebras using rational homotopy
g≥0 Σg• L
techniques as in [FT]. As a corollary of this, it follows that the surface prod-
L
uct ⊎ is homotopy invariant meaning, that, if M and N are 2-connected compact
manifolds with equal dimensions, and i:M →N is a homotopy equivalence, then
i : ( H (Map(Σg,M)),⊎) →( H (Map(Σg,N)),⊎) is an isomorphism
∗ g≥0 • g≥0 •
of algebras.
L L
Section4isdevotedtotheHochschildhomologyof(differentialgraded)symmet-
ric algebras (S(V),d) and a Hochschild-Kostant-Rosenberg type theorem. Recall
that classically (when d = 0), this theorem states that the Hochschild homology
HH (S(V),S(V)),thoughtofastheHochschildhomologyoffunctions onthedual
•
space V∗, can be identified with the space of K¨ahler differential Ω , that is the
S(V)
spaceofpolynomialdifferentialformsΩ ∼=Ω•V∗ =S(V)⊗Λ(V)=S(V⊕V[1]).
S(V)
A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT 5
This result and its extension to the case where d 6= 0 are main tools for comput-
ing Hochschild homology in algebra and topology; see [L, BV, FTV1, FTV2] for
instance. NotethatthereisanobviousidentificationΩ =S(H (S1)⊗V). Sim-
S(V) •
ilarly it is shown in [P] that HHS2(S(V),S(V))=S(V ⊕V[2])=S(H (S2)⊗V).
• •
We prove a similar kind of theorem for surfaces. Indeed we show, that for a
surface Σg of genus g, there is an algebra isomorphism
H S(H (Σg)⊗V),dΣg −∼→HHΣg•(S(V),S(V))
• • •
wheredΣg isadiffer(cid:0)entialbuildoutofthe(cid:1)differentialdandthecoalgebrastructure
of H (Σg), see Theorem 4.3.2. It is worthnoticing that dΣg =0 iff d=0. Further,
•
there is a commutative diagram
2g p q
H S(H ( S1)⊗V) //H S(H (Σg)⊗V) //H S(H (S2)⊗V)
• • • • • •
i=1
(cid:0) W (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
∼= ∼= ∼=
HH•W2i=g1S1(S(cid:15)(cid:15) (V),S((VW2i)=d)1S1֒→Σ//gH)•H•Σg(S((cid:15)(cid:15)V),S(V))(Σg։S2)//•HH•S2(S(V(cid:15)(cid:15) ),S(V))
wherethehorizontalmapsp,qarethealgebrahomomorphismsrespectivelyinduced
2g
bythehomologymapsH ( S1)⊗V)→H (Σg)andH (Σg)→H (S2)obtained
• • • •
i=1
by the obvious inclusion andWsurjection of spaces and the lower maps are obtained
by functoriality of Hochschild homology.
The main idea that is used to prove this result is, that if X is a space ob-
tained by attaching various spaces along attaching maps, then the Hochschild ho-
mology HHX(A,A) can be computed by the Hochschild homology of the various
•
pieces and the attaching maps via derived tensor products. For instance, a genus
g surface can be obtained by suitably gluing a square along its bondary on a bou-
quet of circles from which we deduce that there is an isomorphism CHΣg(A,N)∼=
•
CHW2i=g1S1(A,N) ⊗L CHI2(A,A)foranyA-moduleN;seeSection4.1. With
• •
CHS1(A,A)
•
thistoolinhand,wereducetheproofofthemaintheoremtoappropriatestatements
about CHW2i=g1S1(A,M) and CHI2(A,A) for which the usual Hochschild-Kostant-
• •
Rosenberg Theorem and contractibility of the square I2 may be used. .
Further, the Hochschild-Kostant-Rosenberg type theorem for surfaces allows us
toexplicitlycomputethesurfaceproductforoddspheresS2n+1 andaLiegroupG.
The idea is that in both cases, the differential graded algebras Ω•S2n+1 and Ω•G,
are quasi-isomorphic to symmetric algebras with zero differentials; see Examples
4.4.6 and 4.4.7 in Section 4.4.
Acknowledgments. We would like to thank the Max-Planck Institut fu¨r Mathe-
matik in Bonn, where this project initiated. Also, we thank the Einstein Chair of
the City University ofNew York,andthe UniversityPierreetMarieCurie ofParis
6 for their invitations. The last two authors are partially supported by the NSF
FRG grant DMS-0757245.
Conventions.
- Inthispaperweuseacohomologicalgradingforallour(co)homologygroups
and graded spaces, even when using a subscript to denote the grading. In
6 G.GINOT,T.TRADLER,ANDM.ZEINALIAN
particular, all differentials are of degree +1, of the form d:Ai →Ai+1 and
the homology groups H (X) of a space X are concentrated in non-positive
i
degree (unless otherwise stated).
- We follow the Koszul-Quillen sign convention: “Whenever something of
degree p is moved past something of degree q the sign (−1)pq accrues”,
see [Q]. In particular, we will often write “±” in the case that the sign
is obtained by a permutation of elements of various degrees, following the
Koszul-Quillen sign rule.
2. Chen’s iterated integrals for Hochschild complexes over
simplicial sets
2.1. Higher Hochschild chain complex over simplicial sets. In this section,
we define Chen’s iterated integral map for any simplicial set Y, and give explicit
versions of it in the three examples of the circle S1, the sphere S2 and the torus
T, and combinations of those. In the next section, we will prove that (under suit-
ableconnectivity conditions)this givesinfacta quasi-isomorphismto the cochains
of the corresponding mapping space MY = Map(Y,M). We start by recalling
the Hochschild chain complex of an differential graded, associative, commutative
algebra A over a simplicial set Y from T. Pirashvili, see [P].
•
Definition 2.1.1. Denote by ∆ the category whose objects are the ordered sets
[k]={0,1,...,k},andmorphismsf :[k]→[l]arenon-decreasingmapsf(i)≥f(j)
fori>j. Inparticular,wehavethemorphismsδ :[k−1]→[k],i=0,...,k,which
i
are injections, that miss i, and we have surjections σ : [k+1] → [k],i = 0,...,k,
j
which send j and j+1 to j.
A finite simplicial set Y is, by definition a contravariant functor from ∆ to the
•
category of finite sets Sets, or written as a formula, Y : ∆op → Sets. Denote by
•
Y =Y ([k]), and call its elements simplicies. The image of δ under Y is denoted
k • i •
by d := Y (δ ) : Y → Y , for i = 0,...,k, and is called the ith face. Similarly,
i • i k k−1
s :=Y (σ ):Y →Y , for i=0,...,k, is called the ith degeneracy. An element
i • i k k+1
in Y is called a degenerate simplex, if it is in the image of some s , otherwise it is
k i
called non-degenerate.
We will mainly be interested in pointed finite simplicial sets. These are defined
to be contravariant functors into the category Sets of pointed finite sets, Y :
∗ •
∆op → Sets . In this case, each Y = Y ([k]) has a preferred element called the
∗ k •
basepoint, and all differentials d and degeneracies s preserve this basepoint.
i i
Furthermore, we may extend these definitions to define simplicial sets, respec-
tivelypointedsimplicialsets,byallowingthetargetofY tobeany(notnecessarily
•
finite) set.
Now, a morphism of (finite or not, pointed or not) simplicial sets is a natural
transformation of functors f : X → Y . Note that f is given by a sequence of
• • • •
maps f :X →Y (preservingthe basepointin the pointed case),which commute
k k k
with the faces f d = d f , and degeneracies f s = s f for all k ≥ 0 and
k i i k+1 k+1 i i k
i=0,...,k.
Definition 2.1.2. Let Y :∆op →Sets be a finite pointed simplicial set, and for
• ∗
k ≥ 0, we set y := #Y −1, where #Y denotes the cardinal of the set Y . Fur-
k k k k
thermore,let(A= Ai,d,·)be a differentialgraded,associative,commutative
i∈Z
algebra, and (M = Mi,d ) a differential graded module over A (viewed as
Li∈Z M
L
A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT 7
a symmetric bimodule). Then, the Hochschild chain complex of A with values in
M over Y• is defined as CH•Y•(A,M):= n∈ZCHnY•(A,M), where
CHY•(A,M):= L(M ⊗A⊗yk)
n n+k
k≥0
M
is givenby a sum of elements of totaldegree n+k. In orderto define a differential
DonCH•Y•(A,M),wedefinemorphismsdi :Yk →Yk−1,fori=0,...,k asfollows.
Firstnote,thatforanymapf :Y →Y ofpointedsets,andform⊗a ⊗···⊗a ∈
k l 1 yk
M ⊗A⊗yk, we denote by f∗ :M ⊗A⊗yk →M ⊗A⊗yl,
(2.1) f (m⊗a ⊗···⊗a )=(−1)ǫ·n⊗b ⊗···⊗b ,
∗ 1 yk 1 yl
where b = a (or b = 1 if f−1(j) = ∅) for j = 0,...,y , and n =
j i∈f−1(j) i j l
m· a . The signǫinequation(2.1)isdeterminedbythe
i∈f−1(baQsepoint),i6=basepoint i
usualKoszulsignruleof(−1)|x|·|y| wheneverxmovesacrossy. Inparticular,there
Q
are induced boundaries (d ) : CHY•(A,M) → CHY• (A,M) and degeneracies
i ∗ k k−1
(s ) :CHY•(A,M)→CHY• (A,M), which we denote by abuse of notationagain
j ∗ k k+1
by di and sj. Using these, the differential D : CH•Y•(A,A) → CH•Y•(A,A) is
defined by letting D(a ⊗a ⊗···⊗a ) be equal to
0 1 yk
yk k
(−1)k+ǫia ⊗···⊗d(a )⊗···⊗a + (−1)id (a ⊗···⊗a ),
0 i yk i 0 yk
i=0 i=0
X X
whereǫi isagaingivenbytheKoszulsignrule,i.e.,(−1)ǫi =(−1)|a0|+···+|ai−1|. The
simplicial conditions on d imply that D2 =0.
i
More generally, if Y : ∆op → Sets is a finite (not necessarily pointed) sim-
•
plicial set, we may still define CH•Y•(A,A) := n∈ZCHnY•(A,A) via the same
formula as above, CHnY•(A,A) := k≥0(A⊗A⊗Lyk)n+k. Formula (2.1) again in-
duces boundaries d and degeneracies s , which produce a differential D of square
i i
L
zero on CHY•(A,A) as above.
•
Remark 2.1.3. Note thatdue to our gradingconvention,if Ais nongraded(that
is, concentrated in degree 0), then HHY•(A,A) is concentrated in non positive
•
degrees. In particular our grading is opposite of the one in [L].
Note that the equation (2.1) also makes sense for any map of (pointed) sets
f : X → Y where X and Y are simplicial (pointed) sets. Since A is graded
k k • •
commutative and M symmetric, (f ◦ g)∗ = f∗ ◦ g∗, hence Y• 7→ CHY•(A,M)
is a functor from the category of finite pointed simplicial sets to the category of
simplicial k-vector spaces, see [P]. If M = A, CHY•(A,A) is a functor from the
category of finite simplicial sets to the category of simplicial k-vector spaces.
Definition 2.1.4. Denote by D the subspace of CHYk+1(A,M) spanned by
k+1 •
all degenerate objects, D = Im((s ) )+···+Im((s ) ). It is well-known ([L,
k+1 0 ∗ k ∗
1.6.4,1.6.5]),thattheDk+1 formanacyclicsubcomplexD• ofCH•Y•(A,M),which
therefore implies that the projection CH•Y•(A,M) → CH•Y•(A,M)/D• is a quasi-
isomorphism. We denote this quotient by NHY•(A,M), and call it the normalized
•
Hochschild complex of A and M with respect to Y .
•
The tensor product of differential graded commutative algebras is naturally a
differential graded commutative algebra. Thus, for any finite simplicial set Y ,
•
8 G.GINOT,T.TRADLER,ANDM.ZEINALIAN
CHY•(A,A) is a simplicial differential graded commutative algebra, and further
•
CHY•(A,M) is a (simplicial) module over CHY•(A,A).
• •
Definition 2.1.5. Let X ,Y , and Z be simplicialsets, andlet f :Z →X and
• • • • • •
g : Z → Y be maps of simplicial sets. We define the wedge W = X ∪ Y of
• • • • • Z• •
X and Y along Z as the simplicial space given by W = (X ∪Y )/ ∼, where
• • • k k k
∼ identifies f (z) = g (z) for all z ∈ Z . The face maps are defined as dW•(x) =
k k k i
dX•(x),dW•(y) = dY•(y) and the degeneracies are sW•(x) = sX•(x),sW•(y) =
i i i i i i
sY•(y) for any x∈X ֒→W and y ∈Y ֒→W . It is clear that W is well-defined
i k k k k •
and there are simplicial maps X →i• W and Y →j• W .
• • • •
If X is a pointed simplicial set, then we can make W into a pointed simplicial
• •
set by declaring the basepoint to be the one induced from the inclusion X →W .
• •
(Note that this is in particular the case, when X ,Y ,Z ,f and g are in the
• • • • •
pointed setting.)
Lemma 2.1.6. There is a map of Hochschild chain complexes
CHX•(A,M)⊗ CHY•(A,A)→CHW•(A,M).
• CH•Z•(A,A) • •
If Z injects into either Z →f• X or Z →g• Y , then this map is in fact an isomor-
• • • • •
phism of simplicial vector spaces.
The tensor product in Lemma 2.1.6 is the tensor product of (simplicial) modules
over the simplicial differential graded commutative algebra CHZ•(A,A).
•
Proof. Using the functoriality of the Hochschildchain complex [P], there is a com-
mutative diagram,
(2.2) CHZ•(A,A) f∗ // CHX•(A,A)
• •
g∗ i∗
(cid:15)(cid:15) (cid:15)(cid:15)
CHY•(A,A) j∗ // CHW•(A,A)
• •
which induces the claimed map. If Z injects for example into X , then the tensor
• •
productA⊗xk+1⊗A⊗zk+1A⊗yk+1 is isomorphic to A⊗(xk−zk)+yk+1, whichgives the
isomorphism of the Hochschild complexes. (cid:3)
2.2. Definition of Chen’s iterated integrals. Assume now, that M is a com-
pact, oriented manifold, and denote by Ω = Ω•(M) the space of differential forms
on M. For any simplicial set Y , we now define the space of Chen’s iterated in-
•
tegrals Chen(MY) of the mapping space MY = Map(Y,M), and relate it to the
Hochschild complex over Y from the previous section.
•
Definition 2.2.1. Denote by Y :=|Y |= ∆•×Y /∼ the geometric realization
• •
of Y . Furthermore, let S (Y) be the simplicial set associated to Y, i.e. S (Y) :=
• • k
`
Map(∆k,Y) is the mapping space from the standard k-simplex ∆k = {0 ≤ t ≤
1
···≤t ≤ 1} to Y. By adjunction, there is the canonical simplicial map η :Y →
k •
S (Y), which is given for i ∈ Y = {0,...,y } by maps η(i) : ∆k → Y in the
• k k
following way,
η(i)(t ≤···≤t ):=[(t ≤···≤t )×{i}]∈ ∆•×Y /∼ =Y.
1 k 1 k •
Here we identify 0 with the base point of Y . (cid:16)a (cid:17)
k
A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT 9
DenotebyMY =Map(Y,M)thespaceofcontinuousmapsfromY toM,which
aresmoothonthe interiorofeachsimplex Image(η(i))⊂Y. RecallfromChen[C,
Definition 1.2.1], that to give a differentiable structure on MY, we need to specify
a set of plots φ:U →MY, where U ⊂Rn for some n, which include the constant
maps, and are closed under smooth transformations and open coverings. To this
end,wedenotetheadjointofamapφ:U →MY byφ :U×Y →M. Then,define
♯
the plots of MY to consist those maps φ:U →MY, for which φ :U ×Y →M is
♯
continuouson U×Y, and smoothon the restrictionto the interior of eachsimplex
of Y, i.e. φ | is smooth.
♯ U×(simplexofY)◦
Following[C,Definition1.2.2],ap-formω ∈Ωp(MY)onMY isgivenbyap-form
ω ∈Ωp(U) for each plot φ :U → MY, which is invariant with respect to smooth
φ
transformationsofthedomain. Letusrecallfrom[C]thatasmoothtransformation
betweentwoplots φ:U →MY andψ :V →MY is a smoothmapθ :U →V such
that the following diagram
φ♯
U ×Y //M
θ (cid:15)(cid:15) nnnnnnψn♯nnnnnn77
V ×Y
commutes. The invariance of a p-form means that θ∗(ω ) = ω . The differential,
ψ φ
wedge product and pullback of forms are all defined plotwise. We will now define
certain forms on MY, which we call (generalized) iterated integrals.
Tothisend,assumethatφ:U →MY isaplotofMY,andwearegiveny +1=
k
#Y many forms on M, a ,...,a ∈Ω=Ω•(M). We define ρ :=ev◦(φ×id),
k 0 yk φ
(2.3) ρ :U ×∆k φ−×→idMY ×∆k −e→v Myk+1,
φ
where ev is defined as the evaluation map,
(2.4) ev(γ :Y →M,t ≤···≤t )= γ◦η(0) (t ≤···≤t ),
1 k 1 k
γ◦η(1) (t1 ≤(cid:0)·(cid:0)··≤tk),(cid:1)··· , γ◦η(yk) (t1 ≤···≤tk) .
Now,the pullback(ρ )∗(cid:0)(a ⊗···(cid:1)⊗a )∈Ω•(U×∆k)(cid:0),maybei(cid:1)ntegratedalong(cid:1)the
φ 0 yk
fiber ∆k, and is denoted by
a ...a := (ρ )∗(a ⊗···⊗a ) ∈Ω•(U).
0 yk φ 0 yk
(cid:18)ZC (cid:19)φ Z∆k
The resulting p=( deg(a )−k)-form a ...a ∈Ωp(MY) is called the (gen-
i i C 0 yk
eralized)iterated integralofa ,...,a . Here, we usedthe symbol instead of ,
P 0 yk R C
since our notation differs slightly from the usual one, where iterated integrals refer
R R
to the integration over the interior of a path, see also example 2.3.1 below.
The subspace of the space of De Rham forms Ω•(MY) generatedby all iterated
integralsisdenotedbyChen(MY). Inshort,wemaypictureaniteratedintegralas
the pullbackcomposedwiththe integrationalongthe fiber∆k ofa forminMyk+1,
ev
MY ×∆k //Myk+1
R∆k
(cid:15)(cid:15)
MY
10 G.GINOT,T.TRADLER,ANDM.ZEINALIAN
We now use the above definition to obtain a chain map ItY• : CHY•(Ω,Ω) →
•
Chen(MY). In sections 2.5 and 2.4, we will see that ItY• is in fact a quasi-
isomorphism and an algebra map. In detail, for homogeneous elements a ⊗···⊗
0
ayk ∈CH•Y•(Ω,Ω), we define
(2.5) ItY•(a ⊗···⊗a ):= a ...a .
0 yk 0 yk
ZC
Lemma 2.2.2. ItY• :CHY•(Ω,Ω)→Chen(MY) is a chain map.
•
Proof. Since, Ca0...ayk is a ( ideg(ai)−k)-form,ItY• is a degree0linear map.
WeneedtoshowthatItY• respectsthe differentials. Usingthedefinitionstogether
R P
with Stokes theorem for integration along a fiber,
d ω = dω± ω,
(cid:18)Z∆k (cid:19) Z∆k Z∂∆k
we see that for some plot φ:U →MY, d ItY•(a0⊗a1⊗···⊗ayk) φ is equal to
(cid:0) (cid:1)
d (ρ )∗(a ⊗a ⊗···⊗a )
φ 0 1 yk
Z∆k
= d (ρ )∗(a ⊗a ⊗···⊗a ) ± (ρ )∗(a ⊗a ⊗···⊗a )
φ 0 1 yk φ 0 1 yk
Z∆k (cid:16) (cid:17) Z∂(∆k)
yk
= (ρ )∗ a ⊗···⊗d(a )⊗···⊗a
φ 0 j yk
Z∆k j=0
X
k
± (ρ )∗ bj ⊗···⊗bj ,
φ 0 yk−1
Z∆k−1 j=0
X
where, for the boundary component, with t ≤ ··· ≤ t = t ≤ ··· ≤ t , bj is
1 j j+1 k i
the product of all the a ’s for which l satisfies d (l)=j. Thus, we recoverprecisely
l i
ItY•(D(a0⊗a1⊗···⊗ayk)). (cid:3)
Recallalsofrom[C, Definition 1.3.2]thatthe spaceofsmoothp-chainsC (MY)
p
on MY is the space generated by plots of the form σ : ∆p → MY. We de-
note by C (MY) = C (MY), and give it a differential in the usual way.
• p p
It follows from [C, Lemma 1.3.1], that the induced homology is the usual one,
L
H(C (MY)) = H (MY) (that is, the singular homology of MY). The canonical
• •
chainmapΩ•(MY)⊗C (MY)→R,givenby<ω,σ >:= ω ,inducesasimilar
• ∆p σ
chain map on the space of iterated integrals, Chen(MY)⊗C (MY)→R.
R •
Finally, we remark, that the construction is clearly functorial in Y . Thus it
•
extends by limits to the case where Y is non necessarily finite. In particular, it
•
alsoallowsustodefineakindofCheniteratedintegral,ifwestartwithatopological
space and not a simplicial set as follows.
Definition 2.2.3. Let X be a (pointed) topological space. Then we have the
(pointed) simplicial set S (X). By definition the Hochschild chain complex of A
•
over S (X) with value in an A-module N is, in (external) degree k the limit
•
CHSk(X)(A,N):= lim N ⊗A⊗i
•
i+→Sk(X)
