The cohomology ring of free loop
spaces
Luc Menichi
Abstract
Let X be a simply connected space and _ a commutative ring.
Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild
cohomology of the singular chains on the pointed loop space HH*S*(X)
is isomorphic to the free loop space cohomology H*(XS1). We proved
that this isomorphism is compatible with both the cup product on
HH*S*(X) and on H*(XS1). In particular, we explicit the algebra
H*(XS1) when X is a suspended space, a complex projective space or
a finite CW-complex of dimension p such that __1__(p-1)!2 _.
Mathematics Subject Classification. 55P35, 16E40, 55P62, 57T30, 55U10.
Key words and phrases. Hochschild homology, free loop space, cup product,
bar construction, simplicial space, Hopf algebra up to homotopy, Adams-
Hilton model, Sullivan model, Pertubation Lemma.
Research supported by the University of Toronto (NSERC grants RGPIN
8047-98 and OGP000 7885).
1
1 Introduction
Let X be a simply connected CW-complex. We consider the free loop space
1
cohomology algebra of X, H*(XS ), with coefficients in any arbitrary com-
mutative ring _.
1
When _ is a field, the algebra H*(XS ) can sometimes be computed via
spectral sequences. For example, Kuribayashi and Yamaguchi [20], by solving
extensions problems by applications of the Steenrod operations, were able to
1
compute via the Eilenberg-Moore spectral sequence, the algebra H*(XS ) for
some simple spaces.
If A is a commutative algebra, the Hochschild homology of A, HH*(A)
can be endowed with the shuffle product. When _ = Q, Sullivan, inventing
Rational Homotopy, constructed a commutative algebra APL(X), called the
polynomial differential forms [14, x10], and proved with Vigue-Poirrier [32]
that there is a natural isomorphism of graded algebras
1
H*(XS ) ~=HH*(APL(X)):
The theory of minimal Sullivan models gives a technic to compute the algebra
1
HH*(APL(X)) and so H*(XS ).
Denote by HH*(A) the Hochschild cohomology of an algebra A with
coefficients in A_ = Hom (A; _) [23, 1.5.5]. If A has a diagonal : A ! AA,
then HH*(A) is equipped with a cup product. The normalized singular
chains on the pointed loop space S*(X) is a differential graded Hopf algebra.
So the Hochschild cohomology HH*(S*(X)) is naturally a graded algebra.
In 1985, Goodwillie [15], Burghelea and Fiedorowiscz [7] proved that there
is an isomorphism of graded modules
1
HH*((S*(X)) ~=H*(XS ):
The main result of this paper is that this isomorphism of graded modules
is, in fact, an isomorphism of graded algebras (Theorem 3.1). This result
1
gives a general method to compute the algebra H*(XS ) over any commuta-
tive ring:
o Determine the Adams-Hilton model [1] of X, A(X), using its cellular
decomposition.
o Compute the structure of Hopf algebra up to homotopy model on
A(X) (Definition 4.3), knowing the structure of graded Hopf algebra
of H*(X) when _ is a field.
2
Note that, if X is an H-space, our approach has no interest, since simply
1
XS t X x X:
Now, Theorem 4.6 allows us to replace, in the Hochschild cohomology, the
differential graded Hopf algebra S*(X) by the Adams-Hilton model of X,
A(X), equipped with its structure of Hopf algebras up to homotopy.
We face now (section 5) a completely algebraic problem: How to com-
pute the Hochschild cohomology on a Hopf algebra up to homotopy whose
underlying algebra is a tensor algebra.
In section 6, we show that in a simple case, this Hochschild cohomology
reduces to the Hochschild homology of a commutative algebra.
In section 7, we investigated the algebra structure of the free loop space
cohomologyion anyjsuspension X. If H*(X) is _-free of finite type, this alge-
1 * *
bra H* (X)S ) is the Hochschild homology of H (X), HH*(H (X)),
equipped with a product completely determined by the cohomology algebra
of the desuspended space, H*(X). Recall from [22] and [29] that,ievenjwhen
1
_ = Q and X is a wedge of spheres, the cohomology algebra H* (X)S )
is particularly difficult to explicit in terms of generators and relations.
In section 8, we prove thatithe freejloop space cohomology on the complex
1
projective space CP n, H* (CP n)S , is isomorphic as graded algebras to the
Hochschild homology HH*(H*(CP n)) and compute it. Suppose that X is a
finite CW-complex of dimension p such that __1__(p-1)!2 _. Anick [2, dualize
Proposition 8.7(a)], extending Sullivan's result, constructed a commutative
algebra C*(L(X)) weakly equivalent as algebras (in the sense of [13, page
832]) to the singular cochains on X, S*(X). We extend Sullivan and Vigue-
Poirrier result in this new context (Theorem 8.4).
We would like to mention that Ndombol and Thomas [4] have also found,
when _ is a field, a general method to compute the free loop space cohomology
algebra of a simply connected space. We thank S. Halperin, J.-C. Thomas
and M. Vigue for their constant support. The main results of this paper
were exposed in September 1999, at the GDR Topologie algebrique meeting
in Paris Nord.
3
2 Algebraic preliminaries and notation
We work over a commutative ring _. We denote by p_ and ___p_respectively
the kernel and cokernel of the multiplication by p in _.
DGA stands for differential graded algebra, DGC for differential graded
coalgebra, DGH for differential graded Hopf algebra and CDGA for commu-
tative DGA. The denomination "chain" will be restricted to objects with a
non-negative lower degree and "cochain" to those with a non-negative upper
degree.
The degree of an element x is denoted |x|. The suspension of a graded
module V is the graded module sV such that (sV )i+1= Vi. Let_C be an
augmented complex. The kernel of the augmentation is denoted C.
The exterior algebra on an element v is denoted Ev. The free divided
powers algebra on an element v, denoted v, is
o the free graded algebra generated by fli(v); i 2 N*, divided by the
(i + j)! i+j
relations fli(v)flj(v) = _______fl (v), if |v| is even,
i!j!
o and is just Ev when |v| is odd.
The tensor algebra on a graded module V is denoted TAV . The tensor coal-
gebra is denoted TCV . Their common underlying module is simply_denoted
TV . Given a conilpotent coalgebra C then any morphism ' : C ! V lifts
uniquely to a unique morphism : C ! TC V of coaugmented coalgebras.
The formula for is given by
+1X __
i-1 __
(c) = 'i O C (c); c 2 C (2.1)
i=1
__ __i
where i-1C: C ! C is the iterated reduced diagonal of C.
Let_A_be an augmented DGA. Denote d1 be the differential of the complex
A T(sA ) A obtained by tensorization._We denote_the tensor product of
the elements a 2 A, sa1 2 sA , . . . , sak 2 sA and b 2_A by a[sa1| . .|.sak]b.
Let d2 be the differential on the graded module A T(sA ) A defined by:
d2a[sa1| . .|.sak]b(=-1)|a|aa1[sa2| . .|.sak]b
k-1X
+ (-1)"ia[sa1| . .|.saiai+1| . .|.sak]b
i=1
- (-1)"k-1a[sa1| . .|.sak-1]akb;
4
Here "i= |a| + |sa1| + . .+.|sai|.
The_bar resolution of A, denoted B(A; A; A), is the (A; A)-bimodule (A
T(sA )A; d1+d2). The (reduced)_bar construction on A, denoted_B(A),_is the
coaugmented DGC (TC sA ; d1+ d2) whose underlying complex (TsA ; d1+ d2)
coincides with _ A B(A; A; A) A _ [13, x4]. The cyclic bar construction
or Hochschild complex is the complex A_AAop_ B(A; A; A) denoted C(A).
Explicitly C(A) is the complex (A T(sA ); d1 + d2) with d1 obtained by
tensorization and
d2a[sa1| . .|.sak](=-1)|a|aa1[sa2| . .|.sak]
k-1X
+ (-1)"ia[sa1| . .|.saiai+1| . .|.sak]
i=1
- (-1)|sak|"k-1aka[sa1| . .|.sak-1];
The Hochschild homology is the homology of the cyclic bar construction:
HH*(A) := H*(C(A)):
The Hochschild cohomology is the graded module
HH*(A) := H*(Hom (A;A)(B(A; A; A); A_)) = H*(C(A)_)
where A_ is considered as an (A; A)-bimodule.
Let A and B be two augmented DGA's, Then we have an Alexander-
Whitney morphism of (A B; A B)-bimodules
AW : B(A B; A B; A B) ! B(A; A; A) B(B; B; B)
where the image of a typical element p q[s(a1 b1)| . .|.s(ak bk)]m n is
Xk
(-1)iip[sa1| . .|.sai]ai+1. .a.km qb1. .b.i[sbi+1| . .|.sbk]n:
i=0
Here [26, 3.7]
Xk j-1X ! Xk !
ii= |q| + |bl||aj| + |q| + |bj||m|
j=1 l=1 j=1
Xk i-1X
+ (j - i)|aj| + (k - i)|m| + |i||q| + (i - j)|bj|:
j=i+1 j=1
5
AW is natural and associative exactly. It is also commutative up to a ho-
motopy of (A B; A B)-bimodules. So we get an Alexander-Whitney map
for the cyclic bar construction
AW : C(A B) ! C(A) C(B):
Consider an augmented DGA K equipped with a morphism of augmented
DGA's : K ! K K. Then the composite
C() AW
: C(K) -! C(K K) -! C(K) C(K)
is a morphism of augmented complex. Therefore HH*(K) has a product.
This the cup product of Cartan and Eilenberg [8, XI.6]. In particular, if K
is a DGH then C(K) is a DGC and HH*(K) is a graded algebra.
3 From the chains on the based loops to the
chains on the free loops
The object of this section is to prove the following theorem linking the chains
on the based loops of a space to the chains on its free loops.
Theorem 3.1 (Compare [15] and [7]) Let X be a path connected pointed
space. Then there is a natural DGC quasi-isomorphism
1
C(S*(X)) '!S*(XS ):
1
In particular, HH*S*(X) ~=H*(XS ) as graded algebras.
Goodwillie [15], Burghelea and Fiedorowiscz [7] proved the isomorphism
1
HH*S*(X) ~=H*(XS ) as graded modules only. To obtain our theorem, we
will follow their proofs. We introduce first some terminology about simplicial
objects.
Let C be a category. A simplicial C-object X is a non-negative graded
object together with morphisms di : Xn ! Xn-1 and si : Xn ! Xn+1, 0
i n satisfying some well-known relations [24, VIII.5.2]. A cosimplicial C-
object is a non-negative graded object together with morphisms ffii: Xn-1 !
Xn and oei : Xn+1 ! Xn, 0 i n satisfying the opposite relations [6,
X.2.1(i)]. If C is a category equipped with a tensor product (more precisely
6
a monoidal category [25, VII.1]) then the tensor product of two simplicial
C-objects X = (Xn; di; si) and Y = (Yn; di; si) is the simplicial C-object
X Y = (Xn Yn; di di; si si).
Consider C to be the category of complexes. To any simplicial C-object
(i.e. simplicial complex) X, we can associate a complex in the category
C (i.e. a complex of complexes) denote KN (X) known as the normalized
chain complex of X [24, VIII.6 for the category of modules]. Consider two
simplicial complexes A and B. We have an Alexander-Whitney morphism
of complexes of complexes [24, VIII.8.6] AW : KN (A B) ! KN (A)
KN (B). Every complex of complexes can be condensated [24, X.9.1] into
a single complex. So by composing the functor KN and the condensation
functor, we have a functor, called the realization and denoted | |, from the
category of simplicial complexes to the category of complexes, equipped with
an Alexander-Whitney morphism of complexes AW : |A B| ! |A| |B|
for any simplicial complexes A and B. In particular, | | induces a functor
from the category of simplicial DGC's to the category of DGC's (Recall that
a simplicial DGC can be defined either as a simplicial object in the category
of DGC's or as a coalgebra in the category of simplicial complexes.).
Given any two topological spaces X and Y , the caligraphic notations
AW : S*(X x Y ) ! S*(X) S*(Y ) and EZ : S*(X) S*(Y ) ! S*(X x Y )
are reserved to the standart normalized Alexander-Whitney map and to the
standart normalized Eilenberg-Zilber map concerning singular chains [10,
VI.12.27-8].
Example 3.2 The cyclic bar construction for differential graded algebras.
Let A be a DGA. Then there is a simplicial complex A defined by nA =
A . . .A = An+1 ,
d0a[a1| . .|.an]= aa1[a2| . .|.an];
dia[a1| . .|.an]= a[a1| . .|.aiai+1| . .|.an] for1 i n - 1;
dna[a1| . .|.an]= ana[a1| . .|.an-1];
sia[a1| . .|.an]= a[a1| . .|.ai|1|ai+1| . .|.an] for0 i n:
The complex |A| is exactly (signs included) C(A) the cyclic bar construction
of A. If K is a DGH then K with the diagonal
(K)
K ----! (K K) ~=K K
7
is a simplicial DGC and |K| is the DGC C(K) denoted in section 2.
Example 3.3 Let G be a topological monoid. The cyclic bar construction of
G [23, 7.3.10] is the simplicial space G defined by nG = Gx. .x.G = Gn+1
and with the same formulas for di and si as in the cyclic bar construction
for DGA's. Since the normalized singular chain functor S* is a functor from
topological spaces to DGC's, S*(G) is a simplicial DGC. Therefore |S*(G)|
is a DGC.
The following Lemma compares the DGC's given by the previous two exam-
ples.
Lemma 3.4 (Compare [15, V.1.2]) Let G be a topological monoid. Then
there is a natural DGC quasi-isomorphism |S*(G)| '!|S*(G)|.
Proof. The Eilenberg-Zilber map EZ : S*(G)n+1 ! S*(Gn+1) is a DGC
quasi-isomorphism and therefore defines a morphism of simplicial DGC's
S*(G) ! S*(G). So, applying the functor | |, we get a DGC quasi-
____
isomorphism. |QED_|
Let n be the standart geometric simplex of dimension n. Let ffii: n-1 !
n and oei: n+1 ! n be the i-th face inclusion and the i-th degeneracy of
n. Then = (n; ffii; oei) is a cosimplicial space [6, X.2.2(i)]. The geometric
realization [27, 11.1] of a simplicial space X is defined as
! OE
a
|X| = Xn x n ~
n2N
where ~ is the equivalence relation generated by
(dix; y) ~ (x; ffiiy); x 2 Xn; y 2 n-1
and (six; y) ~ (x; oeiy); x 2 Xn; y 2 n+1:
Recall that |S*(X)| is a DGC whose diagonal is the composite
|S*()| |AW | AW
|S*(X)| ! |S*(X x X)| ! |S*(X) S*(X)| ! |S*(X)| |S*(X)|
Lemma 3.5 (Compare [5, Theorem 4.1] and [14, 17(a)]) Let X be a sim-
plicial space, good in the sense of [31, A.4]. Then there is a natural DGC
quasi-isomorphism f : |S*(X)| '!S*(|X|).
8
Proof. Let ssn : Xn x n i |X| be the quotient map. The morphism f is
defined as the composite
idSi(Xn)n n EZ n Si+n(ssn)
Si(Xn) -------! Si(Xn) Sn( ) ! Si+n(Xn x ) -----! Sn+i(|X|)
where n 2 Sn(n) is the singular simplex idn . We just have to prove that
f is a DGC morphism. The diagonal map of |X| is equal to the composite
|X| (|proj1|;|proj2|)
|X| ---! |X x X| --------! |X| x |X|
where |X | is the simplicial diagonal of X and where proj1 and proj2 are
the simplicial projections on each factors. So by naturality of f, it suffices
to show that f well behaves with products of simplicial spaces.
Let X and Y be two simplicial spaces. Then S*(X) and S*(Y ) are two
simplicial complexes. So we have an Alexander-Whitney map
|S*(X) S*(Y )| AW!|S*(X)| |S*(Y )|:
Its formula is given by
" #
X M
S*(d"q) S*(dp0): S*(Xn) S*(Yn) -! S*(Xp) S*(Yq)
p+q=n p+q=n
where "dq: Xn ! Xp is the composite dp+1O . .O.dn (d"denotes the "last" face
operator) and dp0: Yn ! Yq is the iteratedPcomposite of d0. InLthe diagram
page 10, there was no space left for sums and direct sums . So we use
the indices p and q with the convention p + q = n and the indices j and k
with the conventions that j + k = i. We use also the maps
"ffiq= ffin O . .O.ffip+1: p ! n;ffip0: q ! n;
"oeq0= oep O . .O.oen-1 : n ! p andoep0: n ! q:
Both the interchange of factors of a tensor product of modules and of a
product of spaces are denoted by o.
Consider the diagram page 10.
9
idn *
* n EZ @
Si(Xn|x Yn)______________//_Si(Xn x Yn) S*
*n( )_________________//Sn+i(Xn x Y@
| *
* @
| | *
* | @
| | *
* | @
| | *
* | @
| idSn() *
* /.-,()*+3 Sn+i(idx@
| *
* @
| | *
* | @
| | *
* | @
| /.-,()*+1 fflffl|*
* EZ fflf@
AW || Si(Xn x Yn) Sn(n x*
* n) ____________//_Sn+i(Xn x Yn @
| *
* @
| | *
* | @
| | *
* | @
| AWAW| *
* /.-,()*+ Sn+i(idxo@
| *
* 4 @
| | *
* | @
| | *
* | @
fflffl|idAW_(n;n)//___________fflffl|*
*______________________________fflf@
Sj(Xn) Sk(Yn) Sj(Xn) Sk(Yn) Sp(n) *
*Sq(n) Sn+i(Xn x n x Y@
TT
| | T*
*TTTTTT | @
| | *
* TTTT | @
| | p *
* TEZEZpO(idoid)TTTT | @
Sj("dq)Sk(dp0) /.-,()*+2Sj("dq)Sk(d0)S*
*p("oeq)Sq(oe0)TTTTTT AW||@
| | *
* TTTTTT | @
| | *
* TTTT | @
fflffl| fflffl|*
* TT)) fflf@
Sj(Xp) Sk(Yq)_______//S (X ) S (Y ) S (p)*
* S (q) /.-,()*+5S (X x n) S@
idpq j p k q pTT *
* q p+j n @
T*
*TT | @
*
* TTTTT | @
*
* TTTTT | @
*
* TTTTT Sp+j("dqx"oeq@
*
* EZEZO(idoid)TTTTTTTT | @
*
* TTTT | @
*
* TTTT)) fflf@
*
* Sp+j(Xp x p) Sq+k@
10
Let's check the commutativity of each subdiagram involved in it.
o 1 commutes obviously since Sn()(n) = (n; n).
o 2 commutes since
" #
M X
Sp("oeq) Sq(oep0)O AW (n; n)= Sp("oeq) Sq(oep0)("ffiq ffip0)
p+q=n p+q=n
X
= p q:
p+q=n
o 3 commutes by naturality of EZ .
o 4 commutes by compatibility of EZ and AW [14, I.4.b)].
o 5 commutes by naturality of (EZ EZ ) O (id o id).
o 6 commutes since
(|proj1|; |proj2|) O ssn = (ssn x ssn) O (id x o x id) O (id x ):
o 7 commutes by naturality of AW .
o 8 does not commute. But the two different maps coming from 8 coincide
on the image of (EZ EZ ) O (id o id) O [id AW (n; n)]. Indeed
this image is embedded in the image of
X
S*(idXn x "ffiq) S*(idYnx ffip0):
p+q=n
Now
ssp O (d"qx "oeq) O (idXn x "ffiq) = ssp O (d"qx idp ) = ssn O (idXn x "*
*ffiq):
We have a similar formula for Yn.
Finally, we have
" #
X
(f f) O S*(d"q) S*(dp0)O |AW | = AW O S*((|proj1|; |proj2|) O f:
p;q
____
|QED_|
11
Lemma 3.6 [23, 7.3.15] Let X be a path connected pointed space. Then
1
there is a natural homotopy equivalence |X| '!XS .
Proof of Theorem 3.1 Applying Lemma 3.4 to the Moore loop space X,
Lemma 3.5 to X and Lemma 3.6 yield to the sequence of DGC quasi-
isomorphisms:
1
CS*(X) = |S*(X)| '!|S*(X)| '!S*(|X|) '!S*(XS ):
____
|QED_|
4 HAH models
In order to compute the algebra structure of HH*S*(X), it is necessary to
replace S*(X) by a smaller Hopf algebra. Let's first remark that the cyclic
bar construction preserves quasi-isomorphisms.
Property 4.1 [23, 5.3.5](Compare [13,_4.3(iii)])_Let f : A ! B be a quasi-
isomorphism of augmented DGA's. If A and B are _-semifree then C(f) :
C(A) '!C(B) is a quasi-isomorphism of complexes.
Let f, g : A ! B be two morphisms of augmented DGA's. A derivation
homotopy_from f to g is a morphism of graded modules of degree +1, h :
A ! B such that dOh+hOd = f -g and h(xy) = h(x)g(y)+(-1)|x|f(x)h(y)
for x; y 2 A. A derivation homotopy from f to g is denoted by h : f t g. We
say that f and g are homotopic if there is a derivation homotopy between
them.
Lemma 4.2 Let f, g : A ! B be two morphisms of augmented DGA's.
If f and g are homotopic then the morphisms of complexes C(f), C(g) :
C(A) ! C(B) are chain homotopic.
Proof. Let h be a derivation homotopy from f to g. By induction on the
wordlength of the cyclic bar construction, construct an explicit chain homo-
____
topy between C(f) and C(g). |QED_|
A Hopf algebra up to homotopy, or HAH, is a DGA K equipped with
two morphisms of DGA's : K ! K K and " : K ! _ such that
(" idK ) O = idK = (idK ") O (counitary exactly), ( 1) O t
(1 ) O (coassociative up to homotopy) and o O t (cocommutative
up to homotopy).
12
Let K, K0be two HAH's. A morphism of augmented DGA's f : K ! K0
is a HAH morphism if f t (f f) (f commutes with the diagonals up
to homotopy).
Definition 4.3Let X be a pointed topological space. A HAH model for X
is a free chain algebra (TA V; @) equipped with a structure of Hopf algebras up
to homotopy and with a HAH quasi-isomorphism : (TA V; @) '!S*(X).
The existence of HAH models for any space is guarantied by the following
two properties.
Property 4.4 [13, 3.1] Let A be a chain algebra. There exists a free chain
algebra (TA V; @) and a quasi-isomorphism of augmented chain algebras
: (TA V; @) -'! A:
Property 4.5 [3, I.7 and II.1.11=II.2.11a)] or [13, 3.6] Consider a quasi-
isomorphism of augmented chain algebras p : A -'! B and a morphism of
augmented chain algebras g from a free chain algebra (TA V; @) to B:
vA::
v
fvv ' p||
v fflffl|
(TA V; @)g__//_B
Then
i) there is a morphism of augmented chain algebras f : (TA V; @) ! A
such that p O f t g,
ii)moreover, any two such morphisms f are homotopic.
Indeed by Property 4.4, we obtain a quasi-isomorphism of augmented DGA's
: (TA V; @) -'! S*(X):
Since (TA V; @) and S*(X) are _-semifree 1, is a quasi-isomorphism.
By Property 4.5 i), we obtain a diagonal TAV for (TA V; @) such that the
___________________________
1We do not assume that _ is a principal ideal domain [13, x2].
13
following diagram of augmented DGA's commutes up to homotopy:
(TA V;O@)_____'_______//S*(X)
TAV OO S*(X)||
fflffl fflffl|
(TA V; @) (TA V; @)'_//_S*(X) S*(X)
Since S*(X) is exactly a DGH and is cocommutative up to homotopy,
by Property 4.5 ii), is counitary, coassociative and cocommutative up to
homotopy. The diagonal can be chosen to be strictly counitary [2, Lemma
5.4].
Theorem 4.6 Let X be a path connected pointed space. Let (TA V; @) be a
HAH model for X. There is a isomorphism of graded algebras
1
HH*(TA V; @) ~=H*(XS ):
Proof. Let : (TA V; @) '!S*(X) denote the HAH quasi-isomorphism. By
Property 4.1, C() is a quasi-isomorphism. According Lemma 4.2,
C(( ) O TAV) t C(S*(X) O ):
So by composing with AW and by applying Theorem 3.1, the quasi-isomorphisms
of chain complexes
C() ' S1
C(TA V; @) ---! CS*(X) ! S*(X )
____
commutes with the diagonals up to chain homotopy. |QED_|
5 A smaller resolution than the bar resolu-
tion
The goal of this section is to replace the huge algebra up to homotopy
C(TA V; @)_ by a smaller in order to be able to compute the algebra HH*(TA V; @*
*).
When _ is a field, Micheline Vigue in [33] gives a small complex ((_ sV ) TV;*
* ffi)
whose homology is the vector space HH*(TA V; @). In fact, in this section,
we show that over any commutative ring _, this complex ((_ sV ) TV; ffi)
is a strong deformation retract of C(TA V; @).
14
Definition 5.1Let (Y; d) be a complex. A complex (X; @) is a strong de-
formation retract of (Y; d) if there exist two morphisms of complexes r :
(X; @) ,! (Y; d), f : (Y; d) i (X; @) and a chain homotopy : (Y; d) ! (Y; d)
such that fr = idX and rf - idY = d + d. The map f is called the
projection and the map r is called the inclusion.
We first consider the case where the differential @ on (TA V; @) is just
obtained by tensorization of the differential of a complex V and is so therefore
homogeneous by wordlength.
Consider the tensor algebra TAV on a complex_V_. Define the augmenta-
tion on TAV such that the augmentation ideal TA V is
T+V = i1 V i:
The bar resolution B(TA V ; TAV ; TAV ) contains a subcomplex (TV (_
sV ) TV; d1 + d2), since
d2(a sv b) = (-1)|a|(av b - a vb):
Proposition 5.2 [23, Proposition 3.1.2] The (TA V; TAV )-bimodule (TV
(_ sV ) TV; d1+ d2) is a strong deformation retract of the bar resolution
B(TA V ; TAV ; TAV ).
Proof. Define the projection f : B(TA V ; TAV ; TAV ) i TV (_sV )TV
on its components fn : TV (sT+V )n TV ! TV (_ sV ) TV :
The map f0 : TV TV ! TV TV is the identity map.
We define f1 : TV sT+V TV ! TV sV TV by
Xn
f1(a[sv1. .v.n]b) = (-1)|v1...vi-1|av1. .v.i-1 svi vi+1. .v.nb
i=1
for a; b 2 TV , v1; . .;.vn 2 V and n 2 N*.
For n 2, fn is the zero map. An easy calculation shows that d2f1 =
f0d2. Since f1(a[sa1a2]b) = f1(a[sa1]a2b) + (-1)|a1|f1(aa1[sa2]b), f1d2 = 0.
Therefore f commutes with d2 and is a morphism of complexes.
Of course, fr = idTV (_sV )TV . The components
n : TV (sT+V )n TV ! TV (sT+V )n+1 TV
15
of the chain homotopy are defined by:
0 = 0;
n(a[sa1| . .|.san-1|sv]b)=0;
n(a[sa1| . .|.san-1|sanv]b)=-(-1)"na[sa1| . .|.san|sv]b
+n(a[sa1| . .|.san]vb)
for a; b 2 TV , v 2 V and a1; . .;.an 2 T+V . Recall that "n = |a| + |sa1| +
. .+.|san|.
By a double induction first on n and then on the wordlength, check that
dn+ n-1d = rfn- id, n 2 N. At the beginning for n = 1, use the formula
____
f1(a[sa1v]b) = f1(a[sa1]vb) + (-1)|a1|aa1 sv b. _____ |QED_|
Consider now an augmented DGA (TA V; @) such that TA V = T+V . The
differential @ decomposes uniquely as a sum d1+d2+. .+.di+. .o.f derivations
satisfying di(V ) TiV = V i. The differential d1 is called the linear part of
d.
To pass from the case @ = d1 to the general case, we'll use the well-known
perturbation Lemma. For an abundant and recent bibliography, see [21] or
[18].
f
Theorem 5.3 (Perturbation Lemma) Let (X; @) o (Y; d) be a strong
r
deformation retract of chain complexes satisfying f = 0, r = 0 and
2 = 0. Suppose moreover that this strong deformation retract is filtered:
there exist on X and on Y increasing filtrations bounded below preserved by
@, d, f, r and . Consider a filtration lowering linear map t : Y ! Y
of degree -1 such that d + t is a new differential on Y (Such t is called a
perturbation). Then
X
@1 = @ + f(t)k-1tr;
k>0
X
r1 = r + (t)kr;
k>0
X
f1 = f + f(t)k;
k>0
X
1 = + (t)k
k>0
16
f1
are well defined linear maps and (X; @1 ) o (Y; d + t) 1 is a strong
r1
deformation retract.
By applying the Perturbation Lemma to Proposition 5.2 we rediscover
Theorem 5.4 [33, Theoreme 1.4] Let (TA V; d) be a chain algebra. Suppose
that V is a graded module concentrated in degree greater or equal than one.
Define the linear map of degree +1
S : TV TV ! TV sV TV
Xn
a v1. .v.n 7! (-1)|av1...vi-1|av1. .v.i-1 svi vi+1. .v.n
i=1
Consider the chain complex (TV (_ sV ) TV; D) where
D|TV TV = d; D|TV sV TV = "d1+ d2;
"d1(a sv b) = da sv b - S(a dv):b - (-1)|av|a sv db
and d2(a sv b) = (-1)|a|(av b - a vb) fora; b 2 TV; v 2 V:
Then (TV (_ sV ) TV; D) is a strong deformation retract of the (TA V; TAV )-
bimodule B(TA V; TAV; TAV ).
Proof. By Proposition 5.2 (TV (_ sV ) TV; d1 + d2) is a strong de-
formation retract of B ((TA V; d1); (TA V; d1); (TAwV;hd1))ere d1 denotes the
linear part of d. The anhilation conditions are satisfied:
1(a[sv]b) = 0 and 0 = 0, therefore r = 0.
The projection f is null on TV (sT+V )2 TV and 0 = 0. Therefore
fn = 0 for n 2 N.
Since n+1n(a[sa1| . .|.sanv]b) = n+1n(a[sa1| . .|.san]vb), by induc-
tion on wordlength n+1n = 0 for n 1.
Let n 2 Z. An element a[sa1| . .|.san]b is said to have a filtration degree
-n if and only if the sum of the wordlengths of a, a1, . . . , an and b is
greater or equal than n. The filtrations are bounded below since V = V1 .
The maps r, f, , d1 and d2 respect wordlengths. Therefore the strong
deformation retract is filtered. Define the perturbation t to be equal to
the differential of B ((TA V; d); (TA V; d); (TAmV;id))nus the differential of
B ((TA V; d1); (TA V; d1); (TA.V;Sd1))ince d2 = d - d1 increases wordlength
by 1 at least, t is filtration lowering.
17
So finally we can apply the Perturbation Lemma and (TV (_ sV ) TV; @1 )
is a strong deformation retract of B ((TA V; d); (TA V; d); (TA.V; d))
The composite tn maps TV (sT+V )n TV into TV (sT+V )n+1
TV , 0 is null. Therefore f(t)k = 0 for k 1. So f = f1 (The projection is
unchanged) and D := @1 = @ + ftr = d2+ fd1r where d1 is the linear part
of the the differential of B ((TA V; d); (TA V; d); (TA.V;Sd))et "d1:= fd1r.
"d1(a sv b) = da sv b - (-1)|a|f1(a sdv b) - (-1)|av|a sv db:
S(a dv):b = (-1)|a|f1(a sdv b):
____
|QED_|
Corollary 5.5 [33, Theoreme 1.5] Let (TA V; d) be a chain algebra such that
V = V1 . Define the linear map of degree +1
__
S : TV TV ! sV TV
Xn
v1: :v:n a 7! (-1)|v1:::vi-1||vi:::vna|svi vi+1: :v:nav1: :v:i-1
i=1
Consider the complex ((_ sV ) TV; ffi)where
ffi|TV = d;
__
ffi(sv a) = (-1)|a||v|1 av - 1 va + (-1)|sv|sv da - S(dv a):
Then ((_ sV ) TV; ffi)is a strong deformation retract of C(TA V; d).
Proof. The linear maps
(_ sV ) TV ! (TV (_ sV ) TV ) TAV TAV opTV
__v b 7! 1 __v 1 b
(TV (_ sV ) TV ) TAV TAV opTV ! (_ sV ) TV
_v|+|a0|+|b|)_0
a __v a0 b 7! (-1)|a|(| v a ba
are inverse. The strong deformation retract given by Theorem 5.4 is com-
patible with the structure of (TA V; TAV )-bimodule on B(TA V ; TAV ; TAV ).
To obtain the Corollary, we should tensor it by TV TAV TAV op- and then
18
permute TV and (_ sV ). But it is equivalent and shorter to tensor by
____
- TAV TAV opTV and use the previous isomorphisms. |QED_|
Suppose that (TA V; @) is a HAH model of a path connected space X.
Using the inclusion r1 and the projection f = f1 of the strong deformation
retract given by Corollary 5.5, it is now easy to transport the diagonal of
C(TA V; @), denoted C(TAV;@), to ((_ sV ) TV; ffi). Define the diagonal
of ((_ sV ) TV; ffi)simply as the composite (f f) O C(TAV;@)O r1 .
Now ((_ sV ) TV; ffi)_is an algebra up to homotopy whose homology is
isomorphic to HH*(TA V; @) as graded algebras. This algebra up to homotopy
is the smallest that computes in general the cohomology algebra of the free
1
loop space H*(XS ). But the formula for the diagonal of ((_ sV ) TV; ffi)
is very complicated: it involves in particular the formula of the inclusion r1
given by the Perturbation Lemma.
We will now limit ourself to two important cases where the HAH structure
on (TA V; @) is simple:
o The differential @ is the sum d1 + d2 of only its linear part d1 and
its quadratic part d2. The elements of V are primitive: TAV is a
primitively generated Hopf algebra. This will be the object of Section 6.
o The differential @ is equal to its_linear part d1 (hypothesis_of Proposi-
tion 5.2). The reduced diagonal of TAV is such that (V ) V V .
This will be the object of Section 7.
6 The isomorphism between HH* (C) and HH*(C_ )
__
Let C be a coaugmented DGC. Denote by C the kernel of the counit._The co-
bar construction on C, denoted C, is the augmented DGA TA(s-1C ); d1 + d2
where d1 and d2 are the unique derivations determined by
d1s-1c = -s-1dc and
X __
d2s-1c = (-1)|xi|s-1xi s-1yi; c 2 C
i
__ X
where the reduced diagonal c = xi yi. We follow the sign convention
i
of [12].
19
Theorem 6.1 [19, Theorem A][17, Theorem II] Consider a coaugmented
DGC C _-free of finite type such that C = _ C2 . Then there is a natural
isomorphism of graded modules
HH*(C) ~=HH*(C_):
We give again the proof of Jones and McCleary since we want to check
carefully the signs. We also need to explicit the isomorphism in order to
transport later the algebra structure. We remark that already at the level of
complexes, there is a quasi-isomorphism from C(C)_ to C(C_).
Before giving the proof, we need to give the signs convention used in
this paper. Let f : V ! V 0and g : W ! W 0be two linear maps then
f g : V W ! V 0 W 0is the linear map given by
(f g)(v w) = (-1)|g||v|f(v) g(w):
Therefore if f0 : V 0! V " and g0: W 0! W " are two other linear maps then
0|0 0
(f0 g0) O (f g) = (-1)|f||g(f O g ) (f O g):
Let ' : M ! N be a linear map. If f 2 Hom (N; _) then
'_(f) = (-1)|f||'|f O ':
In particular, if (M; d) is a complex, the dual complex is (M_ ; d_). Let
: N ! Q be another linear map. Then ( O ')_ = (-1)|'|||'_ O _.
Proof. We apply Corollary 5.5 when the chain algebra (TA V; d) is the cobar
C. We obtain a strong deformation retract of the cyclic bar construction
C(C) of the form (C C; ffi). The differential ffi is given by
ffia= d a; a 2 C;
ffi(c a)= dc a + (-1)|c|c da
-1 (s-1c)a - (-1)|xi|xi (s-1yi)a
-1c| -1 (|a|+|y |)|s-1x |-1
+(-1)|a||s 1 as c + (-1) i yi ais xi:
__
Therefore (C C; ffi) is the complex (C Ts-1C_; d1 + d2) where_d1 is just
obtained by tensorization and d2 : C (s-1C )n-1 ! C (s-1C )n is the
sum of n + 1 terms ffi0, ffi1, . . . , ffin given by
-1 -1__ n-1
ffi0 = - (C s ) O (s C ) ;
20
__ i-1 -1 -1 -1__ n-1-i
ffii= C (s-1C ) (s s ) O O s (s C ) ; 1 i n - 1
-1__ n-1 -1 -1__ n-1
andffin = C (s C ) s O oC;C(s-1__C)n-1O (s C ) :
__
Let A_denote_the augmented DGA C_. The differential d2 : A (sA )n !
A (sA )n-1 is also the sum of n + 1 terms d0, d1, . . . , dn (compare to
Example 3.2) given by
-1 __ n-1
d0 = O (A s ) (sA ) ;
__ i-1 -1 -1 __ n-i-1
di= A (sA ) s O O (s s ) (sA ) ; 1 i n - 1
__ n-1 __ n-1 -1
and dn = - (sA ) O oA(s__A)n-1 ;AO A (sA ) s :
___ ~= __
The isomorphism : s(C ) ! (s-1C )_ is such that (s-1)_ O = s-1 [16, p.
276]. For any two complexes V and W , the map : V _ W _! (V W )_
given by (f g) = _O(f g) is a morphism of complexes and is associative,
commutative, natural with respect to linear maps of any degree. Therefore
the composite
__ n An _ -1__ _n -1__ n _
A (sA ) -! C (s C ) ! C (s C )
commutes with diand ffi_ifor 0 i n. So finally
~= _
O [A T()] : C(A) ! (C C; ffi)
____
is an isomorphism of complexes. |QED_|
Let V be a graded module. The tensor algebra TAV can be made into a
cocommutative Hopf algebra by requiring the elements of V to be primitive
[34, 0.5 (10)]. We will called the resulting diagonal, the shuffle diagonal.
Dually the tensor coalgebra TCV equipped with the shuffle product is a com-
mutative Hopf algebra. The shuffle product is defined by
X
[v1| : :|:vp] . [vp+1| : :|:vp+q] = oe . [v1| : :|:vp+q]
oe
where the sum is taken over the (p; q)-shuffles oe and a permutation oe acts on
[v1| : :|:vp+q] by permuting the factors with appropriate signs [16, Appendix]
or [34, 0.5 (8)]. Suppose that V is _-free of finite type and V = V1 . Then
~=
the map : TC(V _) ! (TA V )_ is an isomorphism of Hopf algebras.
21
Let C_be_a cocommutative coaugmented DGC. Then the cobar C =
(TA (s-1C ); d1 + d2) equipped with the shuffle diagonal is a DGH [34, 0.6
(2)]. Dually,_let A be an augmented CDGA. Consider the multiplication
on A TC(sA ) A obtained by_tensorizing_the multiplication of A and
the shuffle_product of TC (sA ). Then the bar resolution of A, B(A; A; A) =
(A TC(sA_) A; d1 + d2) is a CDGA. The cyclic bar construction C(A) =
(A TC(sA ); d1+ d2) is also a CDGA. Therefore the Hochschild homology of
a CDGA A, HH*(A), has a natural structure of commutative graded_algebra
[23, 4.2.7]. The reduced bar construction of A, B(A) = (TC (sA ); d1 + d2) is
a commutative DGH [34, 0.6 (1)].
Theorem 6.2 Under the hypothesis of Theorem 6.1, if C is cocommutative
then the isomorphism HH*(C) ~=HH*(C_) is an isomorphism of commu-
tative graded algebras.
__
Property 6.3 Let K be a graded Hopf algebra. Consider Ker_ the primi-
tive elements of K and the graded coalgebra K TC_(sKer_ )K obtained by
tensorization. Then the canonical map K TC(sKer ) K ! B(K; K; K)
is a morphism of graded coalgebras.
Proof of Theorem 6.2
When restricted to conilpotent coaugmented DGC's, the cobar construc-
tion is a left adjoint functor to the bar construction B [12, Proposition
2.11]. By Formula 2.1, the adjunction map oeC : C '! BC is given by
+1XX __
oeC(c) = [ss-1c1| . .|.ss-1ci+1]; c 2 C
i=0
__i P
where the iterated reduced diagonal c = c1. .c.i+1. We consider now
the inclusion map r1 of the strong deformation retract given by Theorem
5.4 when the chain algebra (TA V; d) is the cobar C. A simple computation
shows that r1 is C oeC C, the tensor product of the identity maps
and the_adjunction_map._ Now_oeC is a morphism of coalgebras, Im oeC
TC(ss-1C ) and s-1C Ker . Therefore by Property 6.3, the coalgebra
C C C obtained by tensorization is a sub DGC of B(C; C; C).
After tensorizing by - CCop C and dualizing, we obtain the natural
DGA quasi-isomorphism
~= _ ' _
C(C_) ! (C C; ffi) C(C) :
22
____
|QED_|
7 The free loop space on a suspension
In this section, we show how to computeithejcohomology algebra of the free
1
loop space on any suspension, H* (X)S . And even better, from the
DGA S*(X) or any DGA weaklyiequivalent,jwe construct an DGA weakly
1
equivalent to the DGA S* (X)S .
First we introduce some terminology. Let C be a coaugmented DGC. The
C __ __
composite C ! C C ,! TA C TAC extends to an unique morphism of
augmented DGA's __ __ __
TA__C: TAC ! TAC TAC :
__
This DGH structure on the tensor algebra TA C is called the Hopf algebra
structure obtained by tensorization_of_the coalgebra C. Dually, let A be an
A
augmented DGA. The composite TC A TCA i A A ! A lifts to an
unique morphism of coaugmented DGC's
__ __ __
TC__A: TCA TCA ! TCA :
__
This DGH structure on the tensor coalgebra TCA is called the Hopf algebra
structure obtained by tensorization of the algebra A. Using formula 2.1, we
see that the product TC__Aof two elements [a1| . .|.ap] and [b1| . .|.bq] admits
the following description:
A sequence oe = ((0; 0) = (x0; y0); (x1; y1); : :;:(xn; yn)d=e(p;fq))ined by
8
><(xi-1+ 1; yi-1) or
(xi; yi) = (xi-1; yi-1+ 1) or
>:
(xi-1+ 1; yi-1+ 1);
is called a step by step path of length n from (0; 0) to (p;_q)._To any step by
n
step path oe of length n, we associate coe= [c1| . .|.cn] 2 A by the rule
8
>:
A(axi byi) if (xi; yi) = (xi-1+ 1; yi-1+ 1), ithstep in diagonal.
23
Then a straightforward computation establishes
X
TC__A([a1| . .|.ap] [b1| . .|.bq]) = coe (7.1)
oe
where the sum is taken over all the step by step paths oe from (0; 0) to (p; q)
and where is the sign obtained with Koszul rule by mixing the a1; . .;.ap
and the b1; . .;.bq. In particular, when the product of A is trivial, the produ*
*ct
TC__Ais the shuffle product considered page 21.
Suppose that the DGC C is _-free of finite type and such that C = _C1 .
Then the map ___
~= ___
: TC(C_ ) ! TA C
is a DGH isomorphism.
The starting observation of this section is the following consequence of
Bott-Samelson Theorem (see [28, 7.1] for details).
Lemma 7.2 Let X be a path connected space.
______
i) Consider the Hopf algebra structure on TAS*(X) obtained by tensoriza-
tion of the coalgebra S*(X). Then there is a natural DGH quasi-
isomorphism ______
TA S*(X) '!S*(X):
ii)Suppose that H*(X) is _-free. Consider the Hopf algebra structure on
TAH+(X) obtained by tensorization on the coalgebra H*(X). Then
there is a HAH quasi-isomorphism
X : TAH+(X) '!S*(X):
We can choose X to be natural in homology and so natural after passing
to homotopy of algebras (Property 4.5 i)).
Definition 7.3[29, 2.2.11] Let R be a graded algebra. Let M be a (R; R)-
bimodule. The graded module RM, product of R and of M equipped with
the multiplication
(r1; m1)(r2; m2) = (r1r2; r1 . m2 + m1 . r2)
is a graded algebra called the trivial extension of R by M.
24
Lemma 7.4 Let C be a chain coalgebra _-free of finite type such that C =
_ C1 . Denote by A the_cochain_algebra_dual of C. Consider the Hopf
algebra structures on TAC and on TCA obtained by tensorization_of_the coal-
gebra_C_and_the algebra A. Define a structure of (TC A; TCA )-bimodule on
s-1A TA by
(s-1a m) . a1: :a:n=- s-1a TC__A(m a1: :a:n)
- (-1)|an|(|m|+|a1:::an-1|)s-1A(a an) TC__A(m a1: :a:n-1)
and
-1a||a :::an|-1
a1: :a:n. (s-1a m) =- (-1)|s 1s a TC__A(a1: :a:n m)
-1a||a :::an|-1
- (-1)|a1|+|s 2s A(a1 a) TC__A(a2: :a:n m)
__ __ __
for a_2_A , m 2 TA and a1: :a:n2 TC A. Consider the cochain complex
(_s-1A )_equipped_with_the trivial_product._Then_the cyclic bar construction
on (_s-1A ), C(_s-1A ) = TCA (s-1A_TA ),_equipped_with the product
of the trivial extension of TC A by s-1A TA is a cochain algebra equipped
with a natural DGA quasi-isomorphism
__ ' ___
C(_ s-1A ) j C(TA C) :
Property 7.5 Let K be a graded Hopf algebra._The diagonal of the coal-
gebra B(K; K; K) restricted to K (_ sK ) K is the (K; K)-linear map
given by
__
[sx] = [sx] [] + [sy] z[] + (-1)|y|[]y [sz] + [] [sx]; x 2 K
__ P
where the reduced diagonal x = y z.
__
Proof of Lemma 7.4_The tensor algebra TAC is equal as DGA to the cobar
on the DGC _ sC with trivial coproduct. Therefore by Theorem 6.1, we
get immediatly a natural quasi-isomorphism of cochain complexes
__ ' ___
C(_ s-1A ) C(TA C) :
But we have to remember how this morphism_decomposes_in order to trans-
port the multiplication from C(TA C)_ to C(_ s-1A ).
25
__
Since the differential on TAC is only linear, by Proposition 5.2, the canon-
ical inclusion
__ __ __ ' __ __ __
TC (_ sC ) TC ,! B(TA C; TAC ; TAC )
__
is_a quasi-isomorphism_of_complexes._ Since the_reduced_diagonal_of TA C,
TA__Cembeds C into_C C,_by Property_7.5, TC (_ sC ) TC is a sub_
coalgebra of B(TA C;_TAC_;_TAC_). By tensorizing by - TA__CTA__CTAC, we
obtain the DGC (_ sC ) TC with differential given by Corollary 5.5 and
diagonal given by
(sx c) = sx c0 1 c00+ (-1)|c||z|sy c0 1 c00z
0||sz|0 |c0||sx|0
+ (-1)|y|+|c 1 yc sz c" + (-1) 1 c sx c"
__ __ __ P
for x 2 C, c 2 TC and wherePthe reduced diagonal x = y z and the
unreduced diagonal c = c0 c". The canonical inclusion
__ __ ' __
(_ sC ) TC ,! C(TA C)
is a DGC quasi-isomorphism.
In order to dualize (for_details,_review the proof of Theorem 6.1), we see
that the diagonal on (_ sC ) TC is the sum of three terms,
__ __ __
TA__C: TC ! TC TC ;
__ __ __ __ __
1 : sC TC ! sC TC TC
and __ __ __ __ __
2 : sC TC ! TC sC TC :
__
The first term TA__Cis just the diagonal of TAC . The second term 1 is the
composite
__ __ __ -1 __
(sTC TA__C)O(CTC TC i)O(CoC;T__CT__C)O(C TA__C)O(s TC )
__
where i denotes the inclusion C ,! TC . The third term 2 is the composite
__ __ -1 __
(TA__C s TC ) O (i oC;T__C TC ) O (C TA__C) O (s TC ):
__ __
Therefore the product_on (_ s-1A ) TA is_the_sum_of three_terms:_the
product TC__Aof TCA_, the_dual_of 1_: s-1A_ TA TA ! s-1A TA and
the dual of 2 : TA s-1A TA ! s-1A TA .
26
Explicitly the product is given by
(1 m):(1 m0) = 1 TC__A(m m0);
(s-1a m):(1 a1: :a:n) =- s-1a TC__A(m a1: :a:n)
- s-1A(a an) TC__A(m a1: :a:n-1);
(1 a1: :a:n):(s-1a m) =- s-1a TC__A(a1: :a:n m)
- s-1A(a1 a) TC__A(a2: :a:n m)
and
(s-1a m):(s-1a0 m0) = 0:
__ __
for a, a02 A, m, m0and a1: :a:n2 TA and where are the signs obtained
____
exactly by the Koszul sign convention. |QED_|
Theorem 7.6 Let X be a path connected space. If S*(X) is weakly equivalent
as _-free chain coalgebra to a chain coalgebra C _-free of finite type such that
C = _ C1 . Then the singular cochains on the free loop spaces on the
1
suspension of X, S*((X)S ) is weakly_equivalent as cochain algebras to the
cyclic bar construction C(_ s-1C_ ) equipped with the product of the trivial
extension given by Lemma 7.4.
Remark 7.7 If X is a finite simply connected CW-complex then X sat-
isfies the hypothesis of Theorem 7.6. Indeed, the Adams-Hilton model of
X denoted A(X) is a free chain algebra (TA V; @) equipped with a quasi-
isomorphism of augmented chain algebras (TA V; @) !' S*(X) and such
that the complex of indecomposables (V; d1) is the desuspension of the re-
duced cellular chain complex of X. Now the bar construction BS*(X) is
weakly equivalent as _-free chain coalgebras to S*(X). Therefore we can take
C = BA(X).
Example 7.8 Sd, d 1. If d 2 using Remark 7.7, S*(Sd) is weakly DGC
equivalent to BH*(Sd), therefore to H*(Sd). By [13, 7.3], there is a DGH
quasi-isomorphism H*(S1) '!S*(S1). So by Theorem 7.6, as DGA
i 1j
S* (Sd+1)S ~ CH*(Sd+1) = E(s-1v) TCv; d2
where v is an element of degree d. If d is even in _, as DGA
i 1j
S* (Sd+1)S ~ E(s-1v) v; 0
27
and i 1j
H* (Sd+1)S ~=H*(Sd+1) H*(Sd+1)
as graded algebras. We suppose now that d is odd. By dualization, TC v ~=
Ev TC(v2) as graded algebras (James-Toda). So as cochain algebras
i 1j
S* (Sd+1)S ~ E(s-1v) Ev (v2); d2flk(v2) = 2(s-1v)vflk-1(v2); k 1:
i 1j
Therefore, over any commutative ring _, the graded algebra H* (Sd)S
is the module
_ -1 2
_ (2_)+(v2) _v:(v2) _(s-1v):(v2) (___)v:(s v):(v )
2_
equipped with the obvious products. In particular, if 1_22 _, all the products
are trivial.
Example 7.9 Comparaison of CP dand S2 _ . ._.S2d, d 1. The Adams-
Hilton model of CP d is H*(CP d). Therefore (Remark 7.7), S*(CP d) is
weakly DGC equivalent to H*(CP d). So by Theorem 7.6, as cochain algebras
i 1j
S* (CP d)S ~ C(H*(CP d)):
Similarly as cochain algebras
i 1j
S* (S3 _ . ._.S2d+1)S~ C(H*(S3 _ . ._.S2d+1)):
The cochain algebras C(H*(CP d)) and C(H*(S3 _ . ._.S2d+1)) have the
same underlying cochain complex. But the non commutative product on
C(H*(CP d)) given by Lemma 7.4 and using formula 7.1, is far more com-
plicated than the shuffle product on C(H*(S3 _ . ._.S2d+1)).
Proof of Theorem 7.6 By Lemma 7.2 i) and Theorem 3.1 (same proof as
Theorem 4.6), there is a natural DGC quasi-isomorphism
______ ' 1
C(TA S*(X)) ! S*((X)S ):
Since the cyclic bar construction (Property 4.1), dualizing and tensorization_
preserve quasi-isomorphisms between _-free chain complexes, C(TA S*(X))_
28
__
is weakly equivalent as cochain algebras to C(TA C)_. By Lemma 7.4, there
is a DGA quasi-isomorphism
___ ' -1___
C(TA C) ! C(_ s C_ ):
___ 1 ____
Therefore C(_ s-1C_ ) is a weakly DGA equivalent to S*((X)S ). |QED_|
We give now the module structure of the Hochschild homology of a graded
algebra with trivial product. Let V be a graded module. The circular per-
mutation toward right o acts on TnV by
o:[v1| : :|:vn] = (-1)|vn||v1:::vn-1|[vn|v1| : :|:vn-1]:
Define the invariants by
TnV o= {x 2 TnV; o:x = x}; TV o= 1n=0TnV o
and the coinvariants by
TnV 1 n
TnVo = _________________; TVo = n=0T Vo
{x - o:x; x 2 TnV }
Consider the graded algebra (_s-1V ) with trivial product. The Hochschild
homology of (_ s-1V ) is the sum of the invariants and of the coinvariants
of positive length desuspended:
____
HH*(_ s-1V ) = TV o (s-1 1)TVo :
Assume now that V is _-free with basis fi. Then the set of words of length
n on fi, denoted fin, is a basis for TnV = V n . The circular permutation o
acts obviously on fin without sign:
o * v1: :v:n= vnv1: :v:n-1:
Consider the quotient of the set fin by the cyclic group generated by o, .
We denote this quotient set \fin. For any word v1: :v:ndenote by k the
smallest integer such that ok * v1: :v:n= v1: :v:n. Of course the element of
V n , ok:[v1| : :|:vn], is either +[v1| : :|:vn] or -[v1| : :|:vn]. For any v1:*
* :v:n
in fin, denote by sym(v1: :v:n) the element of V n :
k-1X
oi:[v1| : :|:vn]:
i=0
29
As modules, TnV ois the direct sum
(
M _sym(v1: :v:n) if ok:[v1| : :|:vn] = +[v1|;: :|:vn]
_____v1:::vn2\fin2_sym(v1:i:v:n)f ok:[v1| : :|:vn] = -[v1| : :|:vn]
and TnVo is
(
M ________v1:i:v:nf ok:[v1| : :|:vn] =;+[v1| : :|:vn]
_______ k
_____v1:::vn2\fin__2_v1:i:v:nf o :[v1| : :|:vn] = -[v1| : :|:vn]
In particular, if V is concentrated in even degree or 2 = 0 in _ then HH*(_
s-1V ) is _-free.
We suppose now that V is _-free of finite type. Using Roos direct calcula-
tion of the dimension of TnV owhen _ = Q [30, p. 179-80], we see (Compare
[29, Theorems 1.2.1 and 1.2.2]) that the cardinal of
_______ k
v1: :v:n2 \fin such thato :[v1| : :|:vn] = +[v1| : :|:vn] inV _ . ._.V
is given, when 1 6= -1 in _, by
n
1_X X i
"(o ; [v1| : :|:vd|v1| : :|:vd| . .|.v1| : :|:vd]):
n i=1v
1:::vd2fid
Here d is the greatest common divisor of i and n. And the integer
"(oi; [v1| : :|:vd|v1| : :|:vd| . .|.v1| : :|:vd])
is the sign given by the Koszul rule derived from the action of the permu-
tation oi on the element of length n, [v1| : :|:vd|v1| : :|:vd| . .|.v1| : :|:v*
*d]. In
particular, by supposing that V is concentrated in even degree, we see that
the cardinal of \fin is
n
1_X d
(dim V ) :
n i=1
Let X be a path connected space such that H*(X) is _-free of finite type.
The Hopf algebra on TC H+ (X) obtained by tensorization of the algebra
H*(X) is naturally isomorphic as Hopf algebras to the loop space cohomology
H*(X). The Hochschild_homology_of H*(X), HH*(_ s-1H+ (X)) =
TH+ (X)o(s-11)TH+ (X)o is naturallyiisomorphicjas graded modules to
1
the free loop space cohomology H* (X)S . This isomorphism of modules
is in fact an isomorphism of algebras:
30
Theorem 7.10 Assume the above hypothesis. The invariants of TH+ (X)
form a graded subalgebra, denoted TC H+ (X)o, of the loop space cohomol-
ogy H*(X). _Consider_the (TC H+ (X)o; TCH+ (X)o)-bimodule structure
on (s-1 1)TH+ (X)o induced by the structure of (TCH+ (X); TCH+ (X))-
bimodule defined on s-1H+ (X) TH+ (X) in Lemma 7.4.__Then_the_as-
sociated trivial extension of TC H+ (X)o by (s-1 1)TH+ (X)o is naturally
isomorphic as gradedialgebrasjto the free loop space cohomology of the sus-
1
pension of X, H* (X)S .
Proof. Using Lemma 7.2 ii), Theorem 4.6 and Lemma 7.4 with C = H*(X),
we obtain that the cyclic bar construction on H*(X), C(_ s-1H+ (X))
equipped with the product of the trivial extensionigivenjby Lemma 7.4 for
1 ____
A = H*(X) has the same cohomology algebra as H* (X)S . |QED_|
i 1j
Remark 7.11 Theorem 7.10 claims that the algebra H* (X)S depends
i 1j
functorially of the algebra H*(X). But it is useful to remember that H* (X)S
depends functorially of the Hopf algebra structure of the loop space homology
H*(X) = TAH+(X).
For example, if we return to Example 7.9, we obtain the weak equivalences
of cochain algebras
i 1j _
S* (CP d)S ~ C TAH+(CP d) ~ CH*(CP d)
and
i 1j _
S* (S3 _ . ._.S2d+1)S~ C TAH+(S2 _ . ._.S2d) ~ CH*(S3_. ._.S2d+1):
If 1_d!2 _ then
TAH+(CP d) ~=TA H+(S2 _ . ._.S2d)
as graded Hopf algebras. We have the isomorphism of cochain algebras
d _ 2 2d _
C TAH+(CP ) ~=C TAH+(S _ . ._.S ) :
So finally, when 1_d!2 _, we have the isomorphism of graded Hopf algebras
3 2d+1
H*(CP d) ~=H* (S _ . ._.S )
31
and the isomorphism of graded algebras
i 1j i 1j
H* (CP d)S ~=H* (S3 _ . ._.S2d+1)S:
The converses can be proven easily. Denote by x2 the generator of
H2(CP d) ~=H2 (S3 _ . ._.S2d+1) . In H*(CP d), xd26= 0. If d! =
0 in _, xd2= 0 in H* (S3 _ . ._.S2d+1) . Therefore, when d! = 0 in
_, there is no isomorphism of graded algebras between H*(CP d) and
H* (S3 _ . ._.S2d+1) . For any space X such that H*(X; Z) is Z-free
of finite type,
_ * _
H*(X; _) ~=H*(X; Z) Z _ and soH* X; ___ ~=H (X; _) _ ___
d!_ d!_
as graded algebras. So we have the implications:
3 2d+1
H*(CP d; _) ~=H* (S _ . ._.S ); _as graded algebras
_ * 3 2d+1 _
) H*(CP d; ___) ~=H (S _ . ._.S ); ___ as graded algebras
d!_ d!_
_ 1
) ___ is the null ring) __2 _:
d!_ d!
We prove now that if d! hasino inversejin _,ithere is no isomorphismjof
1 * 3 2d+1 S1
graded algebras between H* (CP d)S and H (S _ . ._.S ) . We
have the sequence of isomorphisms of modules [23, 5.3.10]
i 1j _____________
H* (CP d)S ~=HH*(TA (x2; : :;:x2d))~=T(x2; : :;:x2d)o(s1)T(x2; : :;:x2d)o:
i 1 j
Thus H* (CP d)S ; Z is Z-free of finite type. So as for the loop spaces, the
proof for the free loop spaces reduces to the case where d! = 0 in _. For X =
1
CP dor S2_ . ._.S2d, by Serre spectral sequence, the inclusion X ,! XS
induces in cohomology an isomorphism in degree 2 and an monomorphismij
1
in even degree. Therefore if d! = 0 in _, xd26= 0 in H* (CP d)S whereas
i 1j
xd2= 0 in H* (S3 _ . ._.S2d+1)S.
It is worth noting the following particular case of Theorem 7.10. The
Hochschild homology of H*(X), HH*(H*(X)) , has a natural structure
of commutative graded algebra since H*(X) is a CDGA.
32
Corollary 7.12 Let X be a path connected space such that H*(X) isi_-freej
1
of finite type. If the cup product on H*(X) is trivial, then H* (X)S is
naturally isomorphic as graded algebras to HH*(H*(X)) .
This Corollary of Theorem 7.10 is proved more easily by applying just The-
orem 4.6, Lemma 7.2 ii) and Theorem 6.2.
8 The Hochschild homology of a commuta-
tive algebra
If a HAH model of a path connected pointed space X is the cobar construction
on a cocommutative chain coalgebra C _-free of finite type such that C =
_ C2 , by Theorems 4.6 and 6.2, the free loop space cohomology of X,
1
H*(XS ), is isomorphic as graded algebras to the Hochschild homology of
the CDGA C_. In this section, we give various examples of such a space X.
Denote by A the cochain algebra C_. We suppose_now that A is strictly
commutative (i. e. a2 = 0 if a 2 Aodd) and that A is _-semifree. We start by
giving a method as general as possible to compute the Hochschild homology
of A.
Let V be a graded module. The free strictly commutative graded algebra
on V is denoted V . A decomposable Sullivan Model of A is a cochain algebra
of the form (V; d) where V = {V i}i2 is _-free of finite type and d(V )
2 V , equipped with a quasi-isomorphism of cochains algebras (V; d) !'
A. If _ is a principal ideal domain, by Theorem 7.1 of [16], A admits a
minimal Sullivan model. When _ is a field, minimal Sullivan models are the
decomposable ones [16, Remark 7.3 i)].
Anyway, suppose now that we have somehow obtained a decomposable
Sullivan model (V; d) of A over our arbitrary commutative ring _. Propo-
sition 1.9 of [11] (See also [17, p 320-2]) is valid over any commutative ring
_. Therefore consider the multiplication of (V; d):
: (V 0; d) (V "; d) ! (V; d):
By induction on the degree of V , we can construct a factorization of :
i 0 '
(V 0; d) (V "; d) ae (V V " sV; D) i (V; d)
OE
such that
33
(i)D(sv) - (v0- v") 2 (V