A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY
RALPH L. COHEN AND JOHN D.S. JONES
Abstract.Let M be a closed, oriented manifold of dimension d. Let LM be *
*the space of
smooth loops in M. In [2] Chas and Sullivan defined a product on the hom*
*ology H*(LM)
of degree -d. They then investigated other structure that this product i*
*nduces, including a
Lie algebra structure on H*(LM), and an induced product on the S1 equiva*
*riant homology,
HS1*(LM). These algebraic structures, as well as others, came under the *
*general heading
of the "String topologyö f M. In this paper we will describe a realizat*
*ion of the Chas
- Sullivan loop product in terms of a ring spectrum structure on the Tho*
*m spectrum of
a certain virtual bundle over the loop space. We show that this ring spe*
*ctrum structure
extends to an operad action of the the äc ctus operad", originally defin*
*ed by Voronov,
which is equivalent to operad of framed disks in R2. We then describe a*
* cosimplicial
model of this spectrum and, by applying the singular cochain functor to *
*this cosimplicial
spectrum we show that this ring structure can be interpreted as the cup *
*product in the
Hochschild cohomology, HH*(C*(M); C*(M)).
Introduction
Let Md be a closed, oriented d - dimensional manifold, and let LM = C1 (S1, M*
*) be the
space of smooth loops in M. In [2] Chas and Sullivan described an intersection *
*product on
the homology, H*(LM), having total degree -d,
O : Hq(LM) Hr(LM) ! Hq+r-d(LM).
In this paper we show that this product is realized by a geometric structure, n*
*ot on the
loop space itself, but on the Thom spectrum of a certain bundle over LM. We de*
*scribe
this structure both homotopy theoretically and simplicially, and in so doing de*
*scribe the
relationship of the Chas - Sullivan product to the cup product in Hochshild coh*
*omology.
We now make these statements more precise.
Consider the standard parameterization of the circle by the unit interval, ex*
*p : [0, 1] ! S1
defined by exp(t) = e2iit. With respect to this parameterization we can regard*
* a loop
fl 2 LM as a map fl : [0, 1] ! M with fl(0) = fl(1). Consider the evaluation map
___________
Date: July 25, 2001.
The first author was partially supported by a grant from the NSF .
1
2 R.L. COHEN AND J.D.S JONES
ev : LM! M
fl! fl(1).
Now let ' : M ! RN+d be a fixed embedding of M into codimension N Euclidean s*
*pace.
Let N ! M be the N - dimensional normal bundle. Let T h( N ) be the Thom spac*
*e of
this bundle. Recall the famous result of Atiyah [1] that T h( N ) is Spanier -*
* Whitehead
dual to M+ . Said more precisely, let M-TM be the spectrum given by desuspendi*
*ng this
Thom space,
M-TM = -(N+d)T h( N ).
Then if M+ denotes M with a disjoint basepoint, there are maps of spectra
S0 ! M+ ^ M-TM and M+ ^ M-TM ! S0
that establish M-TM as the S - dual of M+ . Said another way, these maps indu*
*ce an
equivalence with the function spectrum M-TM ' Map(M+ , S0). In particular in h*
*omology
we have isomorphisms
Hq(M+ ) ~=H-q(M-TM )
H-q(M-TM ) ~=Hq(M+ )
for all q 2 Z. These duality isomorphism are induced by the compositions
H-q(M-TM ) - -fi-!~H-q+d(M) - -j-! Hq(M)
= ~=
where ø is the Thom isomorphism, and æ is Poincare duality.
Notice by duality, the diagonal map : M ! M x M induces a map of spectra
* : M-TM ^ M-TM ! M-TM
that makes M-TM into a ring spectrum with unit. The unit S0 ! M-TM is the map*
* dual
to the projection M+ ! S0.
Now let T h(ev*( N )) be the Thom space of the pull back bundle ev*( N ) ! LM*
*. Define
the spectrum
LM-TM = -(N+d)T h(ev*( N ))
The goal of this paper is to define and study a product structure on the spectr*
*um LM-TM
which among other properties makes the evaluation map ev : LM-TM ! M-TM a map*
* of
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 3
ring spectra. Here, by abuse of notation, ev is referring the map of Thom spect*
*ra induced
by the evaluation map ev : LM ! M. We will prove the following theorem.
Theorem 1. The spectrum LM-TM is a homotopy commutative ring spectrum with u*
*nit,
whose multiplication
~ : LM-TM ^ LM-TM -! LM-TM
satisfies the following properties.
1. The evaluation map ev : LM-TM ! M-TM is a map of ring spectra.
2. There is a map of ring spectra æ : LM-TM ! 1 ( M+ ) where the target is*
* the
suspension spectrum of the based loop space with a disjoint basepoint. Its*
* ring structure
is induced by the usual product on the based loop space. In homology the *
*map æ* is
given by the composition
æ* : Hq(LM-TM )- fi--!~Hq+dLM --'-! Hq( M)
=
where like above, ø is the Thom isomorphism, and the map ' takes a (q + d)*
* - cycle in
LM and intersects it with the based loop space viewed as a codimension d -*
* submanifold.
3. The ring structure is compatible with the Chas - Sullivan homology product*
* in the sense
that the following diagram commutes:
Hq(LM-TM ) Hr(LM-TM ) ---! Hq+r(LM-TM ^ LM-TM ) --j*-!Hq+r(LM-TM )
? ?
fi?y~= ~=?yfi
Hq+d(LM) Hr+d(LM) - --! Hq+r+d(LM)
O
Remark. In [2] Chas and Sullivan define a regrading of the homology of the loop*
* space
Hq = Hq+d(LM)
with respect to which the product O is of total degree zero. We observe that th*
*e Thom
isomorphism defines an isomorphism
H* ~=H*(LM-TM )
which respects gradings, and by the above theorem is an isomorphism of rings, w*
*here the
ring structure on the right hand side comes from the ring spectrum structure of*
* LM-TM .
Next we show that the ring structure ~ : LM-TM ^ LM-TM ! LM-TM extends to *
*an
operad structure of the äc ctus operad" C defined originally by Voronov. This *
*operad is
4 R.L. COHEN AND J.D.S JONES
homotopy equivalent to the operad of framed little disks in R2. We will recall *
*the definition
of the cactus operad C in section 2, where we will prove the following theorem.
Theorem 2. There are maps of spectra
ik : (Ck)+ ^ k (LM-TM )(k)-! LM-TM
giving LM-TM the structure of a C - operad ring spectrum, compatible with the *
*ring struc-
ture ~ : LM-TM ^ LM-TM ! LM-TM .
Our next result has to do with the simplicial structure of LM-TM , and the r*
*esulting
simplicial description of the product.
Let S1*be the simplicial set decomposition of the circle which has one zero s*
*implex and
one nondegenerate one simplex. In this decomposition there are n + 1 n - simpli*
*ces, all of
which are degenerate for n > 1. We write this as S1n= {n + 1}. Now given any sp*
*ace X,
there is a resulting cosimplicial model for the free loop space, LX = Map(S1, X*
*), X*. The
n - simplices of X* are given by maps
Xn = Map(S1n, X) = Map({n + 1}, X) = Xn+1.
Of course the coface and codegeneracy maps of (LX)* are dual to the face and de*
*generacy
maps of S1*.
Our next result states that there is a similar cosimplicial model for LM-TM .
Theorem 3. For M a closed, oriented manifold, the spectrum LM-TM has the str*
*ucture
of a cosimplicial spectrum which we write as TX*. The k simplices of TX* are gi*
*ven by
TXk = (Xk)+ ^ M-TM .
This cosimplicial structure has the following properties.
1. The ring structure of LM-TM is realized on the (co)simplicial level by pa*
*irings
i j
~k : (Xk)+ ^ M-TM ^ (Xr)+ ^ M-TM ! (Xk+r)+ ^ M-TM
defined by
~k(x1, . .,.xk; u) ^ (y1, . .,.yr; v) = (x1, . .,.xk, y1, . .y.r; *(*
*u ^ v))
where * is the ring structure defined on M-TM described earlier (dual to*
* the diagonal
map : M ! M x M).
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 5
2. Applying the singular chain functor C*(-) to the cosimplicial space TX*, w*
*e get a
natural chain homotopy equivalence between the chains of LM-TM and the Ho*
*chshild
cochain complex
f* : C*(LM-TM ) ~=C*(TX*) ~=CH*(C*(M); C*(M)).
Here the notation CH*(A; A) refers to the Hochshild cochain complex of the*
* form
A ! Hom(A; A) ! . .!.Hom(A n; A) ! Hom(A n+1; A) ! . . .
Furthermore, the pairing on the chains C*(LM-TM ) induced by the ring spe*
*ctrum
structure corresponds via the chain homotopy equivalence f* to the cup pro*
*duct pairing
in CH*(C*(M); C*(M)). This gives ring isomorphisms in homology,
H* ~=H*(LM-TM ) --f*-!~HH*(C*(M); C*(M)).
=
Remark. The fact that the Chas - Sullivan product is realized as the cup produc*
*t in
Hochshild cohomology was also observed by T. Tradler, and will appear in his CU*
*NY
Ph.D thesis.
The paper will be organized as follows. In section 1 we will show how to realiz*
*e the Chas -
Sullivan product using the Pontrjagin - Thom constuction for embedded, finite c*
*odimension
manifolds. We will use this to prove theorem 1. In section 2 we define the cact*
*us operad
C and prove theorem 2. In section 3 we will recall the cosimplicial study of th*
*e loop space
done by the second author in [4], apply the Thom spectrum construction to it an*
*d use it to
prove theorem 3.
The authors are grateful to I. Madsen, J. Morava, G. Segal, D. Sullivan, and *
*U. Tillmann
for helpful conversations regarding this material.
6 R.L. COHEN AND J.D.S JONES
1. The ring structure on LM-TM : The proof of theorem 1
In this section we will describe the ring spectrum structure of the Thom spec*
*trum
LM-TM defined in the introduction, discuss some its properties, and prove theo*
*rem 1.
We begin by restating it.
Theorem 4. The spectrum LM-TM is a homotopy commutative ring spectrum with u*
*nit,
whose multiplication
~ : LM-TM ^ LM-TM -! LM-TM
satisfies the following properties.
1. The evaluation map ev : LM-TM ! M-TM is a map of ring spectra.
2. There is a map of ring spectra æ : LM-TM ! 1 ( M+ ) where the target is*
* the
suspension spectrum of the based loop space with a disjoint basepoint. Its*
* ring structure
is induced by the usual product on the based loop space. In homology the *
*map æ* is
given by the composition
æ* : Hq(LM-TM )- u*--!~Hq+dLM --'-! Hq( M)
=
where u* is the Thom isomorphism, and the map ' takes a (q + d) - cycle in*
* LM and
intersects it with the based loop space viewed as a codimension d - subman*
*ifold.
3. The ring structure is compatible with the Chas - Sullivan homology product*
* in the sense
that the following diagram commutes:
Hq(LM-TM ) Hr(LM-TM ) ---! Hq+r(LM-TM ^ LM-TM ) --~*-!Hq+r(LM-TM )
? ?
u*?y~= ~=?yu*
Hq+d(LM) Hr+d(LM) - --! Hq+r+d(LM)
O
Proof.The multiplicative structure ~ : LM-TM ^ LM-TM ! LM-TM will be defined
using the Pontrjagin - Thom construction. We therefore begin by recalling some *
*properties
of this construction.
Let
e : P k,! Nn+k
be an embedding of closed, oriented manifolds. Let e be a tubular neighborhoo*
*d of of
e(P k), which we identify with the total space of the normal bundle of the embe*
*dding. Let
ø : P k! e [ 1
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 7
be the Pontrjagin - Thom collapse map to the one point compactification, define*
*d by
8
< x ifx 2 e
ø(x) =
: 1 ifx =2 e.
If we identify the compactification with the Thom space of the normal bundle,*
* e[ 1 ~=
P e, then in homology, after applying the Thom isomorphism u* : Hq(P e)-~=--!*
*Hq-n(P k),
we get the üp sh-forward", or ü mkehr" map,
e!: Hq(Nk) - fi*--!Hq(P e)-u*--!~Hq-n(P k).
=
Recall that in the case of the diagonal embedding of a d - dimensional closed*
* oriented
manifold,
: Md ! Md x Md
that the normal bundle is isomorphic to the tangent bundle,
~=T M
so that the Pontrjagin - Thom map is a map ø : M x M ! MTM . Furthermore, the p*
*ush
- forward map in homology,
!: H*(Md x Md) - fi*--!H*(MTM ) --u*-!~H*-d(Md).
=
is simply the intersection product.
Now the Pontrjagin - Thom construction also applies when one has a vector bun*
*dle over
the ambient manifold of an embedding. That is, if one has an embedding e : P k,*
*! Nn+k
as above, and if one has a vector bundle (or virtual bundle) i ! N, then one ob*
*tains a
Pontrjagin - Thom map
ø : i [ 1 ! ('*(i)) [ 1
where ('*(i)) is the tubular neighborhood of the induced embedding of total sp*
*aces '*(i) ,!
i. Now i [ 1 is the Thom space N``, and ('*(i)) [ 1 is the Thom space P '*(``)*
*. eSo the
Pontrjagin map is a map
*(``)
ø : N``! P ' . e
Moreover this construction works when i is a virtual bundle over N as well. I*
*n this case
when i = -E, where E ! N is a k - dimensional vector bundle over N, then the Th*
*om
spectrum N``= N-E is defined as follows. Suppose the bundle E is embedded in a*
* k + M
8 R.L. COHEN AND J.D.S JONES
dimensional trivial bundle, E ,! N x Rk+M . Let E? be the M - dimensional orth*
*ogonal
complement bundle to this embedding E? ! N. Then
?
N-E = -(N+k)NE .
Notice that the Thom isomorhism is of the form u* : Hq(N) ~=Hq-k(N-E ).
In particular, applying the Pontrjagin - Thom construction to the diagonal em*
*bedding
: M ,! M x M, using the virtual bundle -T M x -T M over M x M, we get a map of
Thom spectra,
*(-TMx-TM)
ø : (M x M)-TMx-TM ! MTM
or,
ø : M-TM ^ M-TM ! M-TM .
In homology, this map still realizes the intersection pairing on H*(M), after a*
*pplying the
Thom isomorphism. The map ø defines a ring spectrum structure on M-TM that is *
*well
known to be the Spanier - Whitehead dual of the diagonal map : M ! M x M.
To construct the ring spectrum pairing ~ : LM-TM ^ LM-TM ! LM-TM , we basi*
*cally
üp ll back" the structure ø over the loop space.
To make this precise, let ev x ev : LM x LM ! M x M be the product of the eva*
*luation
maps, and define LM xM LM to be the fiber product, or pull back:
~
LM xM LM ---! LM x LM
? ?
(1.1) ev?y ?yevxev
M ---! M x M.
Notice that LM xM LM is a codimension d submanifold of the infinite dimensional*
* manifold
LM x LM, and can be thought of as
LM xM LM = {(ff, fi) 2 LM x LM such thatff(0) = fi(0)}.
Notice that there is also a natural map from LM xM LM to the loop space LM defi*
*ned by
first applying ff and then fi. That is,
(1.2) fl : LM xM LM ! LM
(ff, fi)! ff * fi
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 9
where
8
< ff(2t) if0 t 1_
ff * fi(t) = 2
: fi(2t - 1) if1_
2 t 1.
Notice that when restricted to the product of the based loop spaces, M x M *
*LM xM
LM, then fl is just the H - space product on the based loop space, M x M ! M.
Notice that by its definition (1.1) the embedding ~ : LM xM LM ,! LM x LM has*
* a
tubular neighborhood ( ~) defined to be the inverse image of the tubular neigh*
*borhood of
the diagonal : M ,! M x M:
( ~) = ev-1( ( )).
Therefore this neighborhood is homeomorphic to the total space of the d - dimen*
*sional
vector bundle given by pulling back the normal bundle of the embedding , which*
* is the
tangent bundle of M:
( ~) ~=ev*( ) = ev*(T M).
Thus there is a Pontrjagin - Thom construction
*(TM)
ø : LM x LM ! LM xM LMev .
As described earlier, we ease the notation by refering to this Thom spectrum as*
* (LM xM
LM)TM . By the naturality of the Pontrjagin - Thom construction, we have a comm*
*utative
diagram of spectra,
LM x LM --fi-! (LM xM LM)TM
? ?
(1.3) ev?y ?yev
M x M ---! MTM
fi
Since, as observed before, ø* : H*(M x M) ! H*(MTM ) ~=H*-d(M) is the interse*
*ction
product, then in homology,
H*(LM x LM) --fi*-!H*((LM xM LM)TM ) --u*-!H*-d(LM xM LM)
can be viewed (as is done in Chas - Sullivan [2]) as taking a cycle in LM x LM,*
* and
"intersecting" it with the codimension d submanifold LM xM LM.
10 R.L. COHEN AND J.D.S JONES
Now observe that the map fl : LM xM LM ! LM defined above (1.2) preserves the
evaluation map. That is, the following diagram commutes:
LM xM LM --fl-! LM
? ?
(1.4) ev?y ?yev
M ---! M
=
Thus fl induces a map of bundles fl : ev*(T M) ! ev*(T M), and therefore a map *
*of Thom
spectra,
fl : (LM xM LM)TM ! LMTM .
Now consider the composition
(1.5) ~~: LM x LM --fi-!(LM xM LM)TM --fl-!LMTM
In homology, the homomorphism
(1.6) H*(LM x LM) - ~~*--!H*(LMTM ) - u*--!~H*-d(LM)
=
takes a cycle in LM xLM, intersects in with the codimension d - submanifold LM *
*xM LM,
and maps it via fl to LM. This is the definition of the Chas - Sullivan product*
* H*(LM).
Now as we did before with the diagonal embedding, we can perform the Pontrjag*
*in -
Thom construction when we pull back the virtual bundle -T M x -T M over LM x LM.
That is, we get a map of Thom spectra
*(-TMx-TM) ev*(TM) ev*( *(-TMx-TM))
ø : (LM x LM)(evxev) -! (LM xM LM) .
But since ev*( *(-T M x -T M)) = ev*(-2T M), we have
ø : LM-TM ^ LM-TM -! (LM xM LM)TM -2TM = (LM xM LM)-TM .
Now by the commutativity of (1.4), fl induces a map of Thom spectra,
fl : (LM xM LM)-TM ! LM-TM
and so we can define the ring structure on the Thom spectrum LM-TM to be the c*
*ompo-
sition
(1.7) ~ : LM-TM ^ LM-TM --fi-! (LM xM LM)-TM - -fl-!LM-TM .
A few properties of this map ~ are now immediately verifiable.
First, ~ is associative. This follows from the naturality of the Pontrjagin -*
* Thom con-
struction, and the fact that the map fl is associative. (Strictly speaking, for*
*mula (1.2) is
A1 - associative as is the usual formula for the product on the based loop spa*
*ce, M.
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 11
However the standard trick of replacing M with öM ore loops" changes the A1 st*
*ructure
to a strictly associative structure. The same technique applies to the map fl. *
*Otherwise,
the spectrum LM-TM will have the structure of an A1 ring spectrum.)
Also, notice tha LM-TM has a unit, ' : S0 ! LM-TM , defined by the composit*
*ion
' : S0--j-! M-TM --ff-!LM-TM
where j is the unit of the ring spectrum structure of M-TM , and oe is the map*
* of Thom
spectra induced by the section of the evaluation map ev : LM ! M defined by vie*
*wing
points in M as constant loops.
Thus ~ : LM-TM ^ LM-TM ! LM-TM defines an associative ring spectrum.
Notice furthermore that in homology, after applying the Thom isomorphism, ~* *
*induces
the same homomorphism as ~~*, and so by (1.6) the following diagram commutes:
Hq-2d(LM-TM ^ LM-TM ) --~*-! Hq-2d(LM-TM )
? ?
(1.8) u*?y~= ~=?yu*
Hq(LM x LM) ---! Hq-d(LM)
O
where O : Hq(LM x LM) ! Hq-d(LM) is the Chas - Sullivan product. This proves pa*
*rt
(3) of theorem 1.
Now by the naturality of the Pontrjagin - Thom construction, the following di*
*agram of
Thom spectra commutes (compare (1.3)
LM-TM ^ LM-TM --~-! LM-TM
? ?
(1.9) evxev?y ?yev
M-TM ^ M-TM ---! M-TM .
fi
Thus the evaluation map ev : LM-TM ! M-TM is a map of ring spectra, which pro*
*ves
part 1 of theorem 1.
We now verify part 2 of theorem 1. Let x0 2 M be a base point, and consider *
*the
following pullback diagram:
M - -j-! LM
? ?
(1.10) p?y ?yev
x0 - --! M.
i
12 R.L. COHEN AND J.D.S JONES
Thus the embedding j : M ,! LM is an embedding of a codimension d submanifold,*
* and
has tubular neighborhood, (j) equal to the inverse image of the tubular neighb*
*orhood of
the inclusion of the basepoint i : x0 ,! M This tubular neighborhood is simply *
*a disk Dd,
and so
(j) ~= M x Dd ~=ffld
where ffld reflects the d dimensional trivial bundle over M. Thus the Thom - P*
*ontrjagin
construction makes sense, and is a map
d d
ø : LM ! Mffl= ( M+ )
where the last space is the d - fold suspension of M with a disjoint basepoint*
*. In homology,
the homomorphism
ø* : Hq(LM) ! Hq( d( M+ ) = Hq-d( M)
denotes the map that is obtained by intersecting a q -cycle in LM with the codi*
*mension d
submanifold M.
By performing the Pontrjagin - Thom construction after pulling back the virtu*
*al bundle
-T M over LM, we get a map of Thom spectra
d j*ev*(-TM)
ø : LM-TM ! ( M)ffl .
But by the commutativity of diagram (1.10), j*ev*(-T M) = p*i*(-T M), which is *
*the
trivial, virtual -d dimensional bundle which we denote ffl-d. So the Pontrjagin*
* - Thom map
is therefore a map of spectra
(1.11) æ : LM-TM --fi-!( M)ffld j*ev*(-TM)= ( M)ffld ffl-d= 1 ( M+ )
where here, like before, 1 ( M+ ) denotes the suspension spectrum of the based*
* loop space
of M with a disjoint basepoint. To complete the proof of theorem 1 we need to p*
*rove that
æ : LM-TM ! 1 ( M+ ) is a map of ring spectra. Toward this end, consider the *
*following
diagram of pull back squares:
~
M x M --'-! LM xM LM ---! LM x LM
?? ? ?
(1.12) y ?yev ?yevxev
x0 ---! M ---! M x M.
i
This gives Pontrjagin - Thom maps
(1.13) LM-TM ^ LM-TM --fi-!(LM xM LM)-TM --fi-! 1 ( M+ ) ^ 1 ( M+ ).
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 13
Notice that by the naturality of the Pontrjagin - Thom construction, the above *
*composition
is equal to
æ ^ æ : LM-TM ^ LM-TM -! 1 ( M+ ) ^ 1 ( M+ ).
Now notice that by the formula for the map fl : LM xM LM ! LM, the following
diagram commutes:
M x M - -'-! LM xM LM
? ?
m?y ?yfl
M - --! LM
where m is the usual multiplication on the based loop space. Pulling back the *
*virtual
bundle -T M over LM, and applying the Pontrjagin - Thom construction, we then g*
*et a
commutative diagram of spectra,
LM-TM ^ LM-TM --fi-!(LM xM LM)-TM - fi--! 1 ( M+ ) ^ 1 ( M+ )
? ?
(1.14) fl?y ?ym
LM-TM - --! 1 ( M+ )
j
Now as observed above (1.13), the top horizontal composition ø O ø is equal to *
*æ ^ æ :
LM-TM ^ LM-TM ! 1 ( M+ ) ^ 1 ( M+ ). Also, fl O ø is, by definition, the *
*ring
structure ~ : LM-TM ^ LM-TM ! LM-TM . Thus the following diagram of spectra
commutes: j^j
LM-TM ^ LM-TM - --! 1 ( M+ ) ^ 1 ( M+ )
? ?
~?y ?ym
LM-TM - --! 1 ( M+ ).
j
*
* __
Thus æ is a map of ring spectra, which completes the proof of theorem 1. *
* |__|
14 R.L. COHEN AND J.D.S JONES
2. The operad structure
In this section we describe an operad structure on the spectrum LM-TM . Thi*
*s is the
äc ctus operad" C. This operad was introduced by Voronov. It has the homotopy o*
*f the
operad of framed little disks in R2. According to Getzler's result [3] this is *
*precisely what
is needed to induce the Batalin - Vilkovisky algebra structure in homology. The*
*refore this
structure can be viewed as a homotopy theoretic realization of the BV - algebra*
* structure
Chas and Sullivan show to exist on the chains of the loop space.
We begin by recalling the definition of cactus operad C. A point in the spac*
*e Ck is a
P k
collection of k oriented circles c1, . .,.ck, with radii ri so that i-1ri= 1.*
* Each circle has
a marked point xi 2 ci. Moreover the circles can intersect each other at a fin*
*ite number
of points (vertices) to create a äc ctus - type configuration". Strictly speaki*
*ng this means
that the dual graph of this configuration is a tree. That is, the äc ctus" (i.e*
* the union of
the circles) must be connected and have no "extra loops". (This is the tree con*
*dition on the
dual graph.) The edges coming into any vertex are also equipped with a cyclic o*
*rdering.
Notice that a cactus (i.e a point in Ck) comes equipped with a well defined m*
*ap from
the unit circle to the boundary of the cactus. That is, the map begins at the m*
*arked point
x1 2 c1, then traverses the circle c1 in the direction of its orientation in a *
*length preserving
manner. When a point of intersection with another circle is reached, the loop t*
*hen traverses
that circle, in the direction of its orientation. This path is continued until*
* it eventually
arrives back at the original basepoint x1 2 c1. Given a cactus c = 2 Ck we let
ffic : S1 ! c1 [ . .[.ck be this loop that traverses the boundary of the cactus.
Notice that Ck has a free action of the symmetric group k defined by permuti*
*ng the
ordering of the circles. The operad action
, : Ck x (Cj1x . .x.Cjk)! Cj
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 15
where j = j1 + . .+.jk is defined as follows. Let c x (c1, . .,.ck) 2 Ck x (Cj1*
*x . .x.Cjk).
Scale the cactus c1 down so that its total radius is the radius r1 of the first*
* component c1
of c 2 Ck. Similarly, scale each of the cacti ci down so that its total radius *
*is the radius ri
of the ithcomponent ciof c. By using the loops ffici(scaled down appropriately)*
* we identify
the component circles ciwith the boundary of the cactis ci. This has the effect*
* of replacing
the k component circles c1, . .,.ck making up the cactus c, by the k cacti, c1,*
* . .,.ck. This
produces a cactus with j1 + . .+.jk = j component circles.
Our goal in this section is to prove theorem 2 of the introduction. We now re*
*state that
theorem.
Theorem 5. There are maps of spectra
ik : (Ck)+ ^ k (LM-TM )(k)-! LM-TM
giving LM-TM the structure of a C - operad ring spectrum, compatible with the *
*ring struc-
ture ~ : LM-TM ^ LM-TM ! LM-TM .
Proof.Given a cactus c = 2 Ck, define LcM to be the mapping space
LcM = Map(c, M).
This space consists of maps from the union c1 [ . .[.ck ! M. The map from the c*
*ircle
ffic : S1 ! c1 [ . .[.ck defines a map from LcM to the loop space,
(2.1) flc : LcM! LM
f! f O ffic
Now LC M can also be viewed as the pullback of an evaluation mapping of the p*
*roduct
(LM)k defined as follows. For each component of the cactus ci, let mi be the nu*
*mber of
points on the circle cithat intersect other components of the cactus. Let mc = *
*m1+. .+.mk.
We define an evaluation map
evc : (LM)k -! (M)mc
as follows. On the circle ci, let y1, . .,.ymi be the points that intersect oth*
*er circles in the
cactus c. Assume that these points are ordered according to the orientation of *
*the circle ci
beginning at the marked point xi 2 ci. Let si : S1 ! ci be the identification o*
*f the unit
circle with ci obtained by scaling down the unit circle so as to have radius ri*
*, and rotating
16 R.L. COHEN AND J.D.S JONES
it so the basepoint 1 2 S1 is mapped to the marked point xi 2 ci. Let u1, . .,.*
*umi be the
points on the unit circle corresponding to y1, . .y.mi2 ci under the map si. De*
*fine
(2.2) evci: LM ! (M)mi
oe! (oe(u1), . .,.oe(umi))
Now define
(2.3) evc = evc1x . .x.evcmi: LM ! (M)m1 x . .x.(M)mi = (M)mc
Now let w1, . .,.wnc 2 c1 [ . .[.ck denote all the points in the cactus that *
*lie in more
than one component. For each such point wi, let ~i be the number of components *
*of the
cactus on which wi lies. We think of ~i as the üm ltiplicityö f the intersecti*
*on point wi.
Notice that we have the relation
ncX
(2.4) ~i= mc.
i=1
The "tree" condition on the dual of the cactus also imposes the following relat*
*ion:
(2.5) mc- nc = k - 1
Now consider the diagonal mapping
c : (M)nc -! (M)mc
defined by the composition
~1x...x ~nc =
c : (M)nc ---------! (M)~1x . .x.(M)~nc - --! (M)mc
where ~i: M ! (M)~i is the ~i -fold diagonal. Observe that the following is a *
*cartesian
pull - back square:
~ c
LcM ---! (LM)k
? ?
(2.6) evint?y ?yevc
(M)nc ---! (M)mc
c
where evint: LcM ! (M)nc evaluates a map f : c ! M at the nc intersection point*
*s,
w1, . .,.wnc 2 c1 [ . .[.ck.
The normal bundle ( c) of the diagonal embedding
~1x...x ~nc =
c : (M)nc ---------! (M)~1x . .x.(M)~nc - --! (M)mc
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 17
is equal to
(~1 - 1)T M x . .x.(~nc- 1)T M -! (M)nc,
where (q)T M denotes the q - fold direct sum of T M with itself as a bundle ove*
*r M. Notice
P n
that since ic=1~i = mc and mc - nc = k - 1 (2.5), then if : M ,! (M)nc is t*
*he full
diagonal map then the pull back of this normal bundle
(2.7) *( ( c)) ~=(k - 1)T M -! M.
Now by pulling back the tubular neighborhood (and the resulting normal bundle*
* ) of c
over LcM, we have a resulting Pontrjagin - Thom to the Thom space,
* ( ( ))
ø : (LM)k -! (LcM)evint c
Notice that by (2.7), we have a bundle isomorphism
ev*int( ( c)) ~=ev*((k - 1)T M)
where ev : LcM ! M is evaluation at x1 2 c = c1 [ . .[.ck. In particular the m*
*ap
flc : LcM ! LM (2.1) is covered by a map of bundles,
ev*int( ( c))-flc--!(k - 1)T M
?? ?
(2.8) y ?y
LcM ---! LM.
flc
Now, like what we did in section 1, let (-T M)k ! (LM)k denote the pull back *
*of the
k -fold exterior product of the virtual bundle -T M ! T M via the k -fold produ*
*ct of the
evaluation map at the basepoint, (ev)k : (LM)k ! (M)k. Then performing the Pont*
*rjagin
- Thom construction on this bundle, we get a map of Thom spectra,
* ( ( )) ~*((-TM)k)
(2.9) ø : (LM-TM )(k)-! (LcM)evint c c
Now by (2.8) we have a map of virtual bundles
ev*int( ( c)) ~ *c((-T M)k)flc---!(k - 1)T M (-k)T-M=--!-T M
?? ?
y ?y
LcM - --! LM.
flc
and therefore a map of Thom spectra
* ( ( )) ~*((-TM)k) -TM
flc : (LcM)evint c c ! LM
18 R.L. COHEN AND J.D.S JONES
We then define the map of spectra i(c) : (LM-TM )(k)! LM-TM to be the composi*
*tion
* ( ( )) ~*((-TM)k) -TM
(2.10) i(c) = flcO ø : (LM-TM )(k)-! (LcM)evint c c ! LM .
This then defines the basic structure map
(2.11) ik : (Ck)+ ^ (LM-TM )(k)-! LM-TM
(c ; u1 ^ . .^.uk)-! i(c)(u1 ^ . .^.uk).
By checking the definiton one sees that these map descend to the orbit of the*
* k - action,
(2.12) ik : (Ck)+ ^ k (LM-TM )(k)-! LM-TM .
Furthermore, checking the definitions, and in particular using the naturality o*
*f the Pontr-
jagin - Thom constructions, one sees that the maps ik fit together compatibly t*
*o define the
C - operad structure of the spectrum LM-TM , as claimed.
We end with two observations. First, let c0 2 C2 be the 2- component cactus c*
*0 = ,
where c1 and c2 are both circles of radii 1=2, both oriented counter - clockwis*
*e, intersecting
at one point. The point of intersection is defined to be the marked point of e*
*ach circle,
x1 = x2 2 c1[c2. Then observe that the induced pairing, i(c0) is equal to the r*
*ing spectrum
multiplication ~ defined in section 1,
i(c0) = ~ : LM-TM ^ LM-TM -! LM-TM .
Thus the operad structure is compatible with the ring spectrum structure.
Second, let oe(c0) 2 C2 denote the same 2 - component cactus as c0, permuted *
*by the
action of the nontrivial element of the symmetric group, oe 2 2. So c1 = . Then
notice that the following diagram of spectra commutes:
``(ff(c0))-TM
LM-TM ^ LM-TM -----! LM
? ?
T?y ?y=
LM-TM ^ LM-TM ---! LM-TM
~
where in this diagram T switches the two factors.
Since the operad space C2 is connected, a path between c0 and oe(c0) yields t*
*he homotopy
commutativity of the ring spectrum structure ~ of LM-TM .
*
*__
|*
*__|
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 19
3. A cosimplicial description of LM and LM-TM and a proof of Theorem 3
In this section we describe a cosimplicial model for the spectrum LM-TM . W*
*e then
describe the ring spectrum structure simplicially. This cosimplicial model will*
* then give a
natural way of relating the singular chains C*(LM-TM ) to the Hochshild cochai*
*n complex
CH*(C*(M), C*(M)), and in particular relate the simplicial model for the ring s*
*tructure of
LM-TM to the cup product structure in this cochain complex. This will allow us*
* to prove
theorem 3.
We begin by reviewing the cosimplicial model of the loop space LX for any spa*
*ce X,
coming from a simplicial decomposition of the circle S1. We refer the reader t*
*o [4] for
details.
Let S1*be the simplicial set decomposition of the circle which has one zero s*
*implex and
one nondegenerate one simplex. In this decomposition there are n + 1 n - simpli*
*ces, all of
which are degenerate for n > 1. We write this as S1n= {n + 1}. Now given any sp*
*ace X,
there is a resulting cosimplicial model for the free loop space, LX, which we c*
*all X*. The
n - simplices of X* are given by maps
Xn = Map(S1n, X) = Map({n + 1}, X) = Xn+1.
Of course the coface and codegeneracy maps of X* are dual to the face and degen*
*eracy
maps of S1*. They are given by the formulas
(3.1) ffii(x0, . .,.xn-1)= (x0, . .,.xi-1, xi, xi, xi+1, . .,.xn-1), 0 i*
* n - 1
ffin(x0, . .,.xn-1)= (x0, x1, . .,.xn-1, x0)
oei(x0, . .,.xn+1)= (x0, . .,.xi, xi+2, . .,.xn+1), 0 i n
Since the geometric realization of S1*is homeomorphic to the circle,
S1 ~=|S1*|,
the öt tal complexö r geometric corealization of X* is homeomorphic to the loo*
*p space,
LX ~=T ot(X*).
This was studied in detail by the second author in [4], and in particular the f*
*ollowing
interpretation of this result was given. For each k, let k be the standard k -*
* simplex:
k = {(x1, . .,.xk) : 0 x1 x2 . . .xk 1}.
20 R.L. COHEN AND J.D.S JONES
Consider the maps
(3.2) fk : k x LX- ! Xk+1
(x1, . .,.xk) x!fl(fl(0), fl(x1), . .,.fl(xk)).
Let ~fk: LX ! Map( k, Xk+1) be the adjoint of fk. Then the following was prov*
*en in
[4].
Q
Theorem 6. Let X be any space, and let f : LX -! k 0Map( k, Xk+1) be the pr*
*oduct
of the maps ~fk. Then f is a homeomorphism onto its image. Furthermore, the ima*
*ge con-
sists of sequences of maps {OEk} which commute with the coface and codegeneracy*
* operators.
We call this space of sequences of maps Map *( *, X*+1) and this is the total s*
*pace of the
cosimplicial space T ot(X*).
By applying singular cochains to the maps fk, one obtains maps
f*k: C*(X) k+1 ! C*-k(LX).
The following was also observed in [4].
Theorem 7. For any space X, the homomorphisms f*k: C*(X) k+1 ! C*-k(LX) fit
together to define a chain map from the Hochshield complex of the cochains of X*
* to the
cochains of the free loop space,
f* : CH*(C*(X)) ! C*(L(X))
which is a chain homotopy equivalence when X is simply connected. Hence it indu*
*ces an
isomorphism in homology
~=
f* : HH*(C*(X)) - --! H*(L(X)).
Remark. Let us clarify some notation. Given an algebra (or differential graded *
*algebra)
A, the the Hochshild complex of A, CH*(A) is a complex of the form
. . .-b--!A n+2 --b-! A n+1 - -b-! . .-.b--!A A --b-! A.
The homology of this algebra is denoted HH*(A). More generally if M is a bimodu*
*le over
A, we denote by CH*(A; M) the Hochshild complex of the form
. .-.-b-! A n+1 M - -b-! A n M - -b-! . .-.b--!A M --b-! M.
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 21
The homology of this complex is denoted HH*(A; M). So in particular if M = A we*
* see
that HH*(A; A) = HH*(A). Dually, we denote by CH*(A; M) the Hochshild cochain
complex of the form
* b* b* b* b*
M - b--! Hom(A; M) ---! . .-.--! Hom(A n; M) ---! Hom(A n+1; M) - --!*
* . . .
Its cohomology is denoted HH*(A; M). By dualizing theorem 6 we obtain the follo*
*wing.
Corollary 8. For any simply connected space X, there is a chain homotopy equiva*
*lence
from the singular chains of the loop space to the Hochshild cochain complex
f* : C*(LX) ! CH*(C*(X); C*(X))
and so an isomorphism in homology,
~=
f* : H*(LX) ---! HH*(C*(X); C*(X)).
Notice that the cochain complex CH*(C*(X); C*(X)) does not in general have a *
*natural
product structure. This is because the coefficients, C*(X), is not in general a*
* ring. Notice
however that the Hochshild complex CH*(C*(X), C*(X)) does in fact have a cup pr*
*oduct
coming from the algebra structure of C*(X). Of course when X is a closed, orien*
*ted manifold
of dimension d, Poincare duality gives a chain homotopy equivalence, C*(X) ~=Cd*
*-*(X),
and so the cochain complex CH*(C*(X); C*(X)) inherits an algebra structure. The*
*refore
by the above corollary, H*(LX) inherits an algebra structure in this case. We w*
*ill see that
this indeed realizes the Chas - Sullivan product. We will show this by showing *
*that when
M is a closed, oriented d - manifold, the Thom spectrum LM-TM inherits a cosim*
*plicial
structure from X* for which the analogue of theorem 7 will yield a natural chai*
*n homotopy
equivalence C*(LM-TM ) ~=CH*(C*(M), C*(M)).
To begin, notice that by the definitions 3.2, the following diagrams commute:
k x LM --fk-!Mk+1
? ?
e?y ?yp1
M ---! M
=
where the left hand vertical map is the evaluation, e((t1, . .,.tk); fl) = fl(0*
*), and the right
hand vertical map is the projection onto the first coordinate. Pulling back the*
* virtual bundle
-T (M) defines a map of virtual bundles
(fk)* : e*(-T M) -! p*1(-T M),
22 R.L. COHEN AND J.D.S JONES
and therefore maps of Thom spectra, (which by abuse of notation we still call f*
*k)
(3.3) fk : ( k)+ ^ LM-TM -! M-TM ^ (Mk)+ .
By taking adjoints, we get a map of spectra,
Q f Q
f : LM-TM - k-k-! kMap(( k)+ ; M-TM ^ (Mk)+ )
where on the right hand side the mapping spaces are maps of unital spectra. Th*
*is map
Q
is just the induced map of Thom spectra of the map f : LX -! k 0Map( k, Xk+1)
described in theorem 6. The following result is induced by theorem 6 by passing*
* to Thom
spectra.
Let TX* be the cosimplicial spectrum defined to be the cosimplicial Thom spec*
*trum of
the cosimplicial virtual bundle -T M. That is, the virtual bundle over the k s*
*implices
Xk = Mk+1 is p*1(-T M). Said more explicitly, TX* is the cosimplicial spectrum *
*whose k -
simplices are the spectrum
TXk = M-TM ^ (Mk)+ .
To describe the coface and codegeneracy maps, consider the maps
~L : M-TM ! M+ ^ M-TM and ~R : M-TM ! M-TM ^ M+
of Thom spectra induced by the diagonal map : M ! M x M. ~L and ~R are the ma*
*ps
of Thom spectra induced by the maps of virtual bundles * : -T M ! p*L(-T M) and
* : -T M ! p*R(-T M), where pL and pR are the projection maps M x M ! M onto t*
*he
left and right coordinates respectively. We then have the following formulas fo*
*r the coface
and codegeneracy maps:
(3.4) ffi0(u; x1, . .,.xk-1)= (vR ; yR , x1, . .,.xk-1)
ffii(u; x1, . .,.xk-1)= (u; x1, . .,.xi-1, xi, xi, xi+1, . .,.xk-1), *
* 1 i k - 1
ffik(u; x1, . .,.xk-1)= (vL; x1, . .,.xk-1, yL),
where ~R (u) = (vR , yR ), ~L(u) = (yL, vL) and
oei(u; x1, . .,.xk+1) = (u; x1, . .,.xi, xi+2, . .,.xk+1), 0 i *
* k
The following result is simply the application of the Thom spectrum functor f*
*or the
virtual bundle -T M to theorem 6.
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 23
Theorem 9. Let M be any closed, d - dimensional manifold, and let
Q f Q
f : LM-TM - k-k-! kMap(( k)+ ; M-TM ^ (Mk)+ )
be the product of the maps of spectra fk, as defined above (3.3). Then f is a h*
*omeomor-
phism onto its image. Furthermore, the image consists of sequences of maps {OE*
*k} which
commute with the coface and codegeneracy operators. We call this space of sequ*
*ences of
maps Map *( *, M-TM ^ (Mk)+ ) and this is the total space of the cosimplicial *
*spectrum
T ot(TX*).
Now for M orientable, recall Atiyah's S - duality between M+ and M-TM descri*
*bed
in the introduction [1]. This defines a chain homotopy equivalence between the*
* cochains
C*(M-TM ) (defined to be the appropriate desuspension of the cochains of the T*
*hom space
of the normal bundle of a fixed embedding M ,! RN ) and the chains of the manif*
*old
C*(M-TM ) ~=C-*(M+ ).
The maps fk : ( k)+ ^ LM-TM -! M-TM ^ (Mk)+ then define maps of cochains,
f*k: C-*(M) C*(M) k ~=C*(M-TM ^ (Mk)+ ) -! C*-k(LM-TM ).
Taking the dual we get a map of chain complexes
(fk)* : C*-k(LM-TM )-! Hom(C*(M) k C-*(M); Z)
~=Hom(C*(M) k; C*(M))
= CHk(C*(M); C*(M))
The following is then a consequence of corollary 8, by passing to Thom spectr*
*a.
Corollary 10. For any oriented, closed manifold M, the chain maps (fk)* fit tog*
*ether to
define a chain homotopy equivalence from the chains of Thom spectrum LM-TM to *
*the
Hochshild cochain complex
f* : C*(LM-TM ) ! CH*(C*(M); C*(M))
and so an isomorphism in homology,
~=
f* : H*(LM-TM )---! HH*(C*(M); C*(M)).
24 R.L. COHEN AND J.D.S JONES
As mentioned in the introduction, the Hochshild cochain complex CH*(C*(M); C**
*(M))
has a cup product structure. Namely, for any algebra A, if
OE 2 CHk(A; A) = Hom(A k; A) and _ 2 CHr(A; A) = Hom(A r; A),
then
OE [ _ 2 CHk+r(A; A) = Hom(A k+r; A)
is defined by
OE [ _(a1 . . .ak ak+1 . . .ak+r) = OE(a1 . . .ak)_(ak+1 . . .ak*
*+r).
For A = C*(M) (where the algebra stucture is the cup product in C*(M)), by taki*
*ng
adjoints, we can think of this as a pairing
(3.5i) j
[ : C*(M) k C*(M) C*(M) r C*(M) -! C*(M) k+r C*(M)
(ff1 . . .ffk `) (fi1 . . .fir-! æ)ff1 . . .ffk fi1 . . .*
*fir ` [ æ.
Now recall that by S - duality, there is a ring spectrum structure
* : M-TM ^ M-TM -! M-TM
dual to the diagonal map : M ! M x M. Passing to chains, * : C*(M-TM )
C*(M-TM ) ! C*(M-TM ), is, with respect to the duality identification C*(M-TM*
* ) ~=
C*(M), therefore the cup product on cochains
* = [ : C*(M) C*(M) ! C*(M).
Thus formula (3.5) for the cup product in Hochshild cochains is therefore reali*
*zed by the
map
h i
(3.6) ~~k,r: M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+-! M-TM ^ (Mk+r)+
(u; x1, . .,.xk) ^ (v; y1, .-.,.yr)! ( *(u, v); x1, . .,.xk,*
* y1, . .,.yr)
The maps ~~k,rdefine maps of the simplices
~~k,r: TXk ^ TXr ! TXk+r
and it is straight forward to check that these maps preserve the coface and cod*
*egeneracy
operators, and so define a map of the geometric corealization (öt tal spectra")
~~: T ot(TX*) ^ T ot(TX*) -! T ot(TX*).
This proves the following.
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 25
Theorem 11. Using the homeomorphism f : LM-TM ~=T ot(TX*) of theorem 9, LM-TM
inherits the structure of a ring spectrum,
~~: LM-TM ^ LM-TM -! LM-TM
which is compatible with the cup product in Hochshild cohomology. That is, with*
* respect to
the chain homotopy equivalence
f* : C*(LM-TM ) ! CH*(C*(M); C*(M))
of corollary 10, the following diagram of chain complexes commutes:
C*(LM-TM ) C*(LM-TM ) --~~-! C*(LM-TM )
? ?
f* f*?y~= ~=?yf*
CH*(C*(M); C*(M)) CH*(C*(M); C*(M)) ---! CH*(C*(M); C*(M)).
[
In view of theorems 9, 10, and 11, theorem 3 will therefore be proven once we*
* prove the
following.
Theorem 12. Let ~ : LM-TM ^ LM-TM ! LM-TM be the ring spectrum structure
defined in section 1. Then the structures ~ and ~~: T ot(TX*) ^ T ot(TX*) ! T o*
*t(TX*) are
compatible in the sense that the following diagram homotopy commutes:
LM-TM ^ LM-TM - ~--! LM-TM
? ?
f^f?y' '?yf
T ot(TX*) ^ T ot(TX*)---! T ot(TX*).
~~
Proof.Let wk,r: k+r ! k x r be the Alexander - Whitney diagonal map. That i*
*s,
for (x1, . .,.xk+r) 2 k+r, then wk,r(x1, . .,.xk+r) = (x1, . .,.xk) x (xk+1, .*
* .,.xk+r) 2
k x r. By the definition of the cosimplicial structure of TX*, to prove the t*
*heorem it
suffices to prove that the following diagrams of spectra commute:
26 R.L. COHEN AND J.D.S JONES
(a)
k+r+^ LM-TM ^ LM-TM - 1^~--! k+r+^ LM-TM
?
wk,r^1?y
?
( k x r)+ ^ LM-TM ^ LM-TM ?yfk+r
?
fk^fr?y
M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ - --! M-TM ^ (Mk+r)+
~~
We verify the commutativity of these diagrams in several steps. First observe t*
*hat the maps
fk x fr : k x r x LM x LM ! Mk+1 x Mr+1
restrict to k x r x LM xM LM to define a map fk,rwhose image is in M x Mk x Mr
making the following diagram commute:
k x r x LM xM LM --,!-! k x r x LM x LM
? ?
fk,r?y ?yfkxfr
M x Mk x Mr ---! Mk+1 x Mr+1
where the bottom horizontal map is the diagonal map:
(m x (x1, . .,.xk) x (y1, . .,.yr))= (m, x1, . .,.xk) x (m, y1, . .,*
*.yr).
By the naturality of the Pontrjagin - Thom construction, we therefore have a co*
*mmutative
diagram of spectra
(b)
( k x r)+ ^ LM-TM ^ LM-TM -1^fi--!( k x r)+ ^ (LM xM LM)-TM
? ?
fk^fr?y ?yfk,r
M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ ---! M-TM ^ (Mk)+ ^ (Mr)+ .
fi
Notice further that by the definition of the maps fn, fk,rand the loop composit*
*ion fl :
LM xM LM ! LM defined in the last section, the following diagram commutes:
A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 27
k+r x (LM xM LM) - 1xfl--! k+r x LM
?
wk,rx1?y
?
k x r x (LM xM LM) ?yfk+r
?
fk,r?y
M x (Mk) x (Mr) - --! Mk+r+1.
=
Passing to Thom spectra this yields the following commutative diagram:
(c)
( k+r)+ ^ (LM xM LM)-TM - 1^fl--!( k+r)+ ^ LM-TM
?
wk,r^1?y
?
( k x r)+ ^ (LM xM LM)-TM ?yfk+r
?
fk,r?y
M-TM ^ (Mk)+ ^ (Mr)+ - --! M-TM ^ Mk+r+.
=
Now consider the following diagram of spectra:
( k+r)+ ^ LM-TM ^ LM-TM - 1^fi--!( k+r)+ ^ (LM xM LM)-TM - 1^fl--!(*
* k+r)+ ^ LM-TM
? ?
wk,r^1?y wk,r^1?y
*
* ?
( k x r)+ ^ LM-TM ^ LM-TM - --! ( k x r)+ ^ (LM xM LM)-TM *
* ?yfk+r
? 1^fi ?
fk^fr?y ?yfk,r
M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ - --! M-TM ^ (Mk+r)+ - --! M-T*
*M ^ (Mk+r)+ .
fi =
Now the top left square in this diagram clearly commutes. The bottom left diag*
*ram is
diagram (b) above, and so it commutes. The right hand rectangle is diagram (c)*
* above,
so it commutes. Therefore the outside of this diagram commutes. Now the top hor*
*izontal
composition is, by definition the map
1 ^ ~ : ( k+r)+ ^ LM-TM ^ LM-TM -! ( k+r)+ ^ LM-TM .
The bottom horizontal map is seen to be
~~: M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ -! M-TM ^ (Mk+r)+
28 R.L. COHEN AND J.D.S JONES
by recalling that the ring multiplication * : M-TM ^ M-TM ! M-TM is the Pon*
*trjagin
- Thom map ø : M-TM ^ M-TM ! M-TM applied to the diagonal embedding : M ,!
M x M.
With these identifications, the outside of this diagram is then diagram (a) a*
*bove. As
observed earlier, the commutativity of diagram (a) proves this theorem, and thi*
*s completes
*
* __
the proof of theorem 2. *
* |__|
References
[1]M.F. Atiyah, Thom complexes , Proc. London Math. Soc. (3) , no. 11 (1961), *
*291-310.
[2]M. Chas and D. Sullivan, String Topology, preprint: math.GT/9911159, (1999).
[3]E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological fi*
*eld theories., Comm. Math.
Phys. 159 no. 2,, (1994), 265-285.
[4]J.D.S Jones, Cyclic homology and equivariant homology Inventionnes Math. 87*
*, no. 2, (1987), 403-423.
[5]T. Tradler, Ph.D thesis, CUNY, to appear.
Dept. of Mathematics, Stanford University, Stanford, California 94305
E-mail address, Cohen: ralph@math.stanford.edu
Department of Mathematics, University of Warwick, Coventry, CV4 7AL England
E-mail address, Jones: jdsj@maths.warwick.ac.uk