The rational homotopy Lie algebra of function


                               spaces
                 Urtzi Buijs and Aniceto Murillo*

       Departamento de Algebra, Geometr'ia y Topolog'ia,

                      Universidad de M'alaga,

                   Ap. 59, 29080 M'alaga, Spain



                            June 17, 2005



                               Abstract

        The rational homotopy Lie algebra of function spaces is fully de-
     scribed.
1    Introduction


Starting with the work of Thom [13] and followed by that of Haefliger [7],

the rational homotopy type of function spaces has been extensively studied.

However, there is no explicit and complete description of the homotopy Lie

algebra structure of such spaces, and only particular cases are known:

   Denote by F(X, Y ) (respec. F*(X, Y )) the space of free (respec. based)

maps from X to Y .  From now on, X and Y  are assumed to be nilpotent

complexes with X finite and Y of finite type over Q. In this way the compo-

nents of both F(X, Y ) and F*(X, Y ) are nilpotent CW-complexes of finite

type over Q and can be rationalized in the classical sense.
______________________________
   *Partially supported by the Ministerio de Ciencia y Tecnolog'ia grant MTM200*
 *4-06262

and by the Junta de Andaluc'ia grant FQM-213.
2000 M.S.C.: 55P62.
Keywords:  Function space, homotopy Lie algebra, Sullivan model, rational homot*
 *opy
theory.



                                   1

   If dim X < conn Y (so that F(X, Y ) is connected) M. Vigu'e [14] showed

that the homotopy Lie algebra ss*F(X, Y )Q (respec.  ss*F*(X, Y )Q ) is iso-

morphic as Lie algebra to H*(X; Q)   ss*(YQ ) (respec. H+ (X; Q)   ss*(YQ )).

Later on, Y. F'elix [4] used essential properties of this homotopy Lie algebra

to show, among other deep results, that the Lusternik Schnirelmann category

of the mentioned component is often infinite.

   Following the Brown-Szczarba approach to the Haefliger model of func-

tion spaces [2], we first obtain a natural description of its rational homotopy

groups in terms of derivations. Then, we give a full and explicit description

of the homotopy Lie algebra structure of F(X, Y )Q and F*(X, Y ). Let us be

more precise:

   Let ( V, d) be a Sullivan model, non necessarily minimal, of Y , i.e., a

cofibrant replacement of a commutative differential graded algebra (CDGA

henceforth) homotopy equivalent to C*(Y ; Q), and let B be a finite dimen-

sional CDGA of the homotopy type of C*(X; Q). Then, there is a model of

F(X; Y ) of the form ( (V   B*), ed) (see next section for proper definitions

and details). By a model of a non connected space, or a map between them,

we mean a Z-graded CDGA, or a CDGA morphism, whose simplicial realiza-

tion has the homotopy type of the singular simplicial approximation of the

chosen space or map.

   Moreover, given a map f :X ! Y ,_there_is a standard procedure [11]

to produce a Sullivan model ( SOE, d) (in fact, the Haefliger model) of the

nilpotent space F(X, Y ; f), the path component of F(X, Y ) containing f.

   Our first_result is that the space of the indecomposables of this model

 SOE, Q(d)  is isomorphic as differential vector spaces to (Der ( V, B; OE), ff*
 *i),

the OE-derivations from  V  to B, being OE:  V  ! B a model of f.  As the

homotopy groups of a nilpotent space are isomorphic to (the dual of) the

homology of the indecomposables of a given model, we obtain:


Theorem 1.   For n   1:


 (i) ssn(F(X, Y ; f)Q ) ~=Hn(Der ( V, B; OE), ffi);


 (ii)ssn(F*(X, Y ; f)Q ) ~=Hn(Der ( V, B+ ; OE), ffi).


   This result, together with the naturality of the process followed, includes

(see Corollary 6) [9, Thm.2.1] and [10, Thm.1] as particular cases.

   Then we proceed to fully and explicitly describe the Lie bracket on

ss*F(X, Y ; f)Q and ss*F*(X, Y ; f)Q in terms of derivation:



                                   2

Theorem 2.   the differential linear map of degree 1


        [ , ]: Der *( V, B; OE)   Der *( V, B; OE) -! Der *( V, B; OE),


defined by

                           X   X
   [', _](v) = (-1)|'|+|_|-1       "ijOE(v1 . .^.vi.^.v.j.v.k.)'(vi)_(vj) ,

                               i6=j

              P
in which dv =    v1 . .v.kand "ij is the sign defined in Lemma 5 below,

induces the Whitehead product in homology. Moreover, the restriction to


      [ , ]: Der *( V, B+ ; OE)   Der *( V, B+ ; OE) -! Der *( V, B+ ; OE),


also induces the Lie bracket in ss*F*(X, Y ; f)Q .


   A similar result gives also an explicit description of higher order White-

head products (see Theorem 12).

   As an immediate application we generalize the result of Vigu'e stated

above: If we denote by *: X ! Y the constant map, ssn(F(X, Y ; *)Q ) (respec.

ssn(F*(X, Y ; *)Q )) is isomorphic as Lie algebras to H*(X; Q) ss*(YQ ) (respec.

H+ (X; Q)   ss*(YQ )).

   Finally, from Theorem 2 we generalize [6] and [8, Thm. 1.2]. For a given

space Y , denote by dlY  the least n (or infinite) for which there is a non

trivial whitehead product of order n in ss*(YQ ) (see x3 for more about this

invariant).


Theorem 3.   If cat0X < dlYQ , then F*(X, Y ; f)Q is an H space for all f.

Equivalently, its rational cohomology algebra is free.


   When f is the constant map we may replace cat0X by the rational cup

length of X recovering the main result in [6].
2    Basics of rational homotopy theory of func-


     tion spaces


We shall be using known results on rational homotopy theory for which [5] is

a very good and standard reference. We now recall some specific facts on the

rational homotopy type of a function space F(X, Y ) starting by its Brown-

Szczarba model.  Consider A = ( V, d) -'! APL (Y ) a Sullivan model, non



                                   3

necessarily minimal, of Y and B -'! APL (X) a quasi-isomorphism with B a

connected finite dimensional CDGA. Let B* = hom (B, Q) be the differential

graded coalgebra dual of B, and consider the Z-graded CDGA  (A   B*)

with the natural differential induced by the one on A and by the dual of

the differential of B.  Now, consider the differential ideal I    (A   B*)

generated by 1   1* - 1 and by the elements of the form

                          X             0
              a1a2   fi -    (-1)|a2||fij|(a1   fij0)(a2   fij00),

                           j

                              P    0     00
a1, a2 2 A, fi 2 B, and  fi =   jfij   fij. Then, the composition


              ae:  (V   B*)    (A   B*) ____-  (A   B*)=I


is an isomorphism of graded algebras [2, Thm.1.2], and therefore, considering

on  (V  B*) the differential ed= ae-1dae, ae is also an isomorphism of CDGA's.

Then, ( (V   B*), ed) is a model of F(X, YQ ) [2, Thm.1.3]. In other words,

S*F(X, YQ ) and the simplicial realization of ( (V   B*), ed) are homotopy

equivalent.

   In order to explicitly determine edon v   fi 2 V   B*, calculate (dv)

fi + (-1)|v|v   dfi and then use the relations which generate the ideal I to

express (dv)   fi as an element of  (V   B*).

   We now explain how to obtain Sullivan models (in fact the Haefliger

models) of the different components of F(X, Y ) [2, 11].  For this we need

some algebraic tools: let ( W, d) be a CDGA in which W is Z-graded, and

let u:  W _____- Q be an augmentation.  Given   = ff .  , ff 2 ( + W 0)

and   2  (W 6=0), we denote by  =u the element u(ff) .  Define a linear

map @ :W 0! W 1as follows:  given w 2 W 0, write dw =  0 +  1 +  2,

with  0 2 ( + W <0) . ( W ),  1 2 ( + W 0) . W 1,  2 2 W 1, and define

@(w) =  1=u +  2.
        ___1                                                  ___1
   call W   a complement_of the image of this map, W 1= @W 0  W  , and
                     1     2  __
define the CDGA  (W     W   , d) as follows:
                ___1
   given w  2  (W     W  2) write dw  =  0 +  1 +  2 +  3, in which
                                                             ___1
 0 2  + W <0.  W ,  1 2  + (@W 0) .  W  0,  2 2 ( + W 0) . ( W     W  2)
          ___1               __
and  3 2  W     W  2. Define dw =  2=u +  3.

   Note that if we have in W a basis {wi} for which dwi 2  W<i, then the
                         ___1        __
image of this basis in  (W     W  2,_d)_makes it a Sullivan model. However,
                                                            ___1        __
even when d is decomposable in  W , d might not be, i.e.,  (W     W  2, d)



                                   4

is not necessarily minimal. This depends on u. In fact,_as we just remarked,

for each w 2 W ,  2=u could be the linear part of dw.

   Next, consider ( (V  B*), ed) the model of the function space F(X, Y ) and

let OE: ( V, d) ! B be a model of a given map f :X ! Y .  The morphism

OE clearly induces a natural augmentation which shall be denoted also by

OE: ( (V   B*), ed) !_Q._Applying the process above to this particular case
                            _______1              __
yields a CDGA ( SOE, d) = ( V   B  *  (V   B*) 2, d) which turns out to be

a Sullivan model of F(X, Y ; f). Moreover, the CDGA morphism

                                  __     _______1           2  __
         !0: ( V, d) ____- ( SOE, d) = ( V   B  *  (V   B*)  , d),

                                                   ________1
!0(v) = v   1* if v 2 V  2, or its projection over V   B*   if v 2 V 1, is a

Sullivan model of the evaluation at the base point !0: F(X, Y ; f) ! Y [11,

Cor. 22].

   While !0(v) could vanish if |v| = 1, when ( V, d) is 1-connected,

                                                __
                       !0: ( V, d) ____- ( SOE, d),


is a KS-extension or a relative Sullivan algebra.  The fibre, which is of the

form                      __      _______
                                         1            2   __
              ( (SOE=V ), d) ~=( (V   B  +  (V   B+ )  ), d),

is a Sullivan model of the fibre of !0: F(X, Y ; f) ! Y , i.e., of F*(X, Y ; f).

   Finally, we set some notation:  for any pair V , B of Z-graded vector

spaces, denote by L(V, B) = {Ln(V, B)}n 0 the graded vector space of its

homomorphisms.  In particular, the dual of a given object (except for B*)

shall be denoted by L(-, Q). There is a natural isomorphism

               ~=                                    |fi|(|v|+|`|)
   : L(V, B) ____-L(V   B*, Q),    (`)(v   fi) = (-1)        fi(`(v)).


   Given a CDGA morphism OE: A  !  B, call (Der(A, B; OE), ffi) the dif-

ferential graded vector space where Dern(A, B; OE) are the OE-derivations of

degree n, i.e., linear maps ` :A* ! B*-n  for which `(ab) = `(a)OE(b) +

(-1)n|b|OE(a)`(b). The differential is defined as usual ffi` = d O ` + (-1)n+1`*
 * O d.

Note that when A =  V , Der( V, B; OE) ~=L(V, B) as graded vector spaces

via the identification ` 7! `|V . We shall denote also by

                                       ~=
                   : Der ( V, B; OE) ____-L(V   B*, Q)


the isomorphism above under this identification.



                                   5

 3    Rational homotopy groups of function spaces


 In this section we prove Theorem_1_and extract some consequences.  Con-

 sider the Sullivan model ( SOE, d) of F(X, Y ; f) and recall that, as for any

 other nilpotent_space, ss*F(X, Y ; f)Q is naturally isomorphic to the dual of

 H* SOE, Q(d) , being SOE~=Q( SOE) =  SOE=( + SOE.  + SOE) the space of inde-

 composables. In other word, it is isomorphic to the dual of the homology of

 the following complex:


                 ________1Q(_d)        2 Q(_d)         3 Q(_d)
         0 ____- V   B*  ____- (V   B*)  ____- (V   B*)  ____- . . .
                                         ________1
    However, as (V   B*)1 = @(V   B*)0   V   B*  , this is exactly the

 homology of this slightly different complex:


               @           1  0 Q(_d)        2 Q(_d)         3 Q(_d)
   (V   B*)0 ____- (V   B*)  ______- (V   B*)  ____- (V   B*)  ____- . . .


    Our main result in this section is that the dual of the complex above is

 isomorphic to (Der ( V, B; OE), ffi) via the map   defined in x2. We prove:


 Theorem 4.   The following commutes:



                                          __ *                   __
                @*                  0   Q(d)                   Q(d)*
 L0(V   B*, Q) oe__ L1(V   B*, Q) oe___________L2(V   B*, Q)  oe_______. . .

        6|                 6|                         6|
     ~= |               ~= |                        ~=|
        |                  |                          |
        |        ffi       |            ffi           |          ffi
Der 0( V, B; OE)oe_Der 1( V, B; OE)oe__________Der 2( V, B; OE)oe_____ . . .

                                                        __                 __
 Proof.Here, for simplicity in the notation, we write Q(d)* instead of L(Q(d), *
 *Q).

 For the same purpose we shall omit signs and write just  . However, a careful

 use of Koszul convention leads to proper sign adjustments.

    We first show that, for n   2, the square

                                    __
                                  Q(d)*
               Ln(V   B*, Q)  oe___________Ln+1(V   B*, Q)

                      6|                           6|
                      |~=|                       ~=||
                      |             ffi            |
               Der n( V, B; OE)oe__________Dern+1 ( V, B; OE)

                                    6

commutes. On one hand, given ` 2 Der n+1( V, B; OE) and v   fi 2 (V   B*)n,


          ( ffi`)(v   fi) =  fi(ffi`(v)) =  fi(d(`(v)))   fi(`(dv)). (*)


On the other hand,

     __*                    __
   Q(d)  ` (v   fi) =   `(Q(d)(v   fi)) =   `({dv   fi}   v   dfi) =


    =   `({dv   fi})   (dfi)(`(v)) =   `({dv   fi})   fi(d(`(v))).  (**)


Here, {dv   fi} denotes the indecomposable part of the image of [dv   fi]

through the morphism:


                ae-1                ________1          2
      A   B*=I ____~=- (V   B*) !  (V   B* )    (V   B* ) =  SOE.



To effectively compute {dv   fi} use first the relations which generates I to

write [dv   fi] as an element of  (V   B*).  Then, cancel all elements of

negative degree and their derivatives, and replace any element of degree zero

by the corresponding scalar via OE. Finally, keep the linear part.

   At the sight of (*) and (**), it will be enough to prove:

Lemma 5.   Given   2  V , fi 2 B* and ` 2 Der *( V, B; OE), ( `)({

fi}) = (-1)|fi|(|`|+|f|)i(`( )).


Proof.Denote by FB :B  B* ____- Q and FB B : (B  B) (B* B*) ____- Q

the maps defined respectively by FB (b   fi) = (-1)|b|fi(b) and FB B (b   b0
                     0|+|b|+|b0| 0  0
fi   fi0) = (-1)|b||b     fi(b)fi (b ).  Then, if ~ is multiplication in B, it *
 *is

easy to see that the following commutes:


                                      ~   1B*
                  (B   B)   B*  __________________-B   B*

                        |                             |
              1B B    4 ||                            |FB|
                        ?|                            ?|
              (B   B)   (B*   B*)  _________________-Q.
                                      F(B B)


To prove the lemma, assume   =  kV  and argue by induction on k.  For

k = 1,   = v 2 V and {v fi} = v fi for which the lemma holds by definition

of  . Assume   =   . v with   2  k-1V . Again, to avoid excessive notation,

we shall not write signs:

                                   7




 fi `(  . v)  =  fi `( )OE(v)   OE( )`(v)

              =   FB (`( )OE(v)   fi)   FB (OE( )`(v)   fi) =

              =   FB B (`( )   OE(v)   4fi)   FB B (OE( )   `(v)   4fi)

              =  (a) + (b).
On the other hand
                                     X
            ( `){  . v   fi} = ( `){     (    fi0j)(v   fi00j)},

                                      j
            P
being 4fi =    jfi0j  fi00j.  By definition of { , }, we may keep only those

summands for which one of the factors is of degree zero.  Hence, the above

equality becomes:
           X                                       X
   ( `){          (    fi0j)(v   fi00j)} + ( `){          (    fi0j)(v   fi00j)}

         | |+|fi0j|=0                            |v|+|fi00j|=0

   = (e) + (f)


Using definition and induction hypothesis we get:
                       X
            (f)  =            ( `){    fi0j}OE(v   fi00j)

                     |v|+|fi00j|=0
                       X
                 =            fi0j`( ) fi00jOE(v)

                     |v|+|fi00j|=0|
                       X
                 =            FB B (`( )   OE(v)   fi0j  fi00j)

                     |v|+|fi00j|=0|

                 =    FB B (`( )   OE(v)   4fi) = (a)


Using repeatedly a similar argument one checks that (b) = (e) and the proof
                                                                         __
is complete.                                                            |__|


Finally, we see that
                                               __ *
                     @*                  0   Q(d)
      L0(V   B*, Q) oe__ L1(V   B*, Q) oe___________L2(V   B*, Q)

             6|                 6|                         6|
          ~= |               ~= |                        ~=|
             |                  |                          |
             |        ffi       |            ffi           |
      Der0( V, B; OE)oe_Der 1( V, B; OE)oe__________Der 2( V, B; OE)


                                   8

commutes.  For it note that given v   fi 2 (V   B*)0, @(v   fi) = {dv

fi} + (-1)|v|v   dfi. Hence, using Lemma 5, and following exactly the above

argument:


(@* O  `)(v   fi) = (-1)|`|+1 `({dv   fi} + (-1)|v|v   dfi) = (  O ffi`)(v   fi*
 *),


which gives the commutativity of the left square. For the right_square,_write
                                                                   1
w 2 (V   B*)1 as a sum x + v   fi, x 2 @(V   B*)0, v   fi 2 V   B*  . Then,

           __ *                   |`|+1      __
   ( 0   Q(d)   O  `)(w)  =   (-1)    ( `) Q(d)(v   fi)

                          =   (-1)|`|+1( `)({dv   fi} + (-1)|v|v   dfi)

                          =   (  O ffi`)(w).

                                                                        __
                                                                       |__|


Proof of Theorem 1.  Part (i) is_immediate from Theorem 4. For (ii) consider

the Sullivan model ( (SOE=V ), d) of F*(X, Y ; f) recalled in the past section,

and observe that ssn F*(X, Y ; f)Q   is then isomorphic to the dual of the

homology of the following complex:


                ________1Q(_d)          2 Q(_d)          3 Q(_d)
        0 ____- V   B+   ____- (V   B+ )  ____- (V   B+ )  ____- . . .


To finish restrict Theorem 1 to the dual of this complex.


   We now check that the above isomorphism is natural and respects the

evaluation map at the base point.  Fix a map f :X ! Y  between nilpo-

tent complexes of finite type over Q and let Z be a finite nilpotent com-
                         '                          _
plex.  Let A = ( W, d) _____'-APL (X) and ( V, d) _____'-APL (Y ) be Sulli-

van models (again non necessarily minimal!)  of X and Y  respectively, let

C ____'-APL (Z) be a quasi-isomorphism with C connected finite dimensional,

and let i :( V, d) ____- ( W, d) be a Sullivan model for f. Define



    , :( (V   C*), ed) ____- ( (W   C*), ed),   ,(v   c) = ae-1[i(v)   c],



being ( (V  C*), ed) and ( (W  C*), ed) the models of F(Z, Y ) and F(Z, X)
                                       ~=
respectively, and ae: ( (W  C*), ed) ____- ( (A C*), d)=I the CDGA isomor-

phism described in x2. In other words, to compute effectively ,(v  c) use the

relations which define I to express i(v)   c as an element of  (V   C*). For



                                   9




                                  P                        P            0
instance, if i(v) = w1w2 and 4c =   ic0i  c00j, ,(v   c) =   i(-1)|w2||ci|(w1

c0i)(w2   c00i).

   Finally, let OE: ( W, d) ! C and OE O i :( V, d) ! C be models of g :Z !

X and f O g :Z ! Y respectively. Then [11, Thm.24], the diagram


                          __       ~,                 __
                 ( SOE=W, d) oe___________ ( SOEOi=V, d)

                      6|                         6|
                      |                          |
                      |                          |
                      |   __       ,~            |   __
                   ( SOE, d)oe______________( SOEOi, d)

                      6||                        6||
                    !0|                          !0|
                      |                          |
                      |                          |
                   ( W, d) oe_________________( V, d)
                                    i


is a Sullivan model of


                                  (f)*
                F*(Z, X; g) _____________-F*(Z, Y ; f O g)

                      |                         |
                      |                         |
                      |                         |
                      ?|         (f)*           ?|
                 F(Z, X; g) _____________-F(Z, Y ; f O g)

                      |                         |
                   !0 ||                        |!0|
                      ?|                        ?|
                     X _______________________-Y.
                                  f


   Hence, in view of Theorem 1, the following, which includes [9, Thm.2.1]

and [10, Thm.1], is an easy exercise:


Corollary 6.  (1) For n   1, ssn(f*)Q :ssnF(Z, X; g)Q ! ssnF(Z, Y ; f O g)Q

is naturally equivalent to


          H(i*): HnDer *( W, C; OE) ____- HnDer *( V, C; OE O i).
                                   10

   (2) Moreover,


                                ssn(f*)Q
              ssnF*(Z, X; g)Q____________-ssnF*(Z, Y ; f O g)Q

                     |                            |
                     |                            |
                     |                            |
                     ?|         ssn(f*)Q          ?|
              ssnF(Z, X; g)Q _____________-ssnF(Z, Y ; f O g)Q

                     |                            |
             ssn(!0)Q||                           |ssn(!0)Q|
                     ?|                           ?|
                 ssn(X)Q _|__|_|__||__|_||____-ssn(Y )Q
                                ssn(f)Q


is equivalent to


                                 H(i*)
       Hn(Der *( W, C+ ; OE))____________-Hn(Der *( V, C+ ; OE O i))

                 |                                   |
                 |                                   |
                 |                                   |
                 ?|             H(i*)                ?|
        Hn(Der *( W, C; OE))_____________-Hn(Der *( V, C; OE O i))

                 |                                   |
           H("*) ||                                  |H("*|)
                 ?|              H(i)                ?|
        Hn(Der *( W, Q; ")) _______________-Hn(Der *( V, Q; ")).





   Note that here (Der *( V, Q; "), ffi)  ~=  (L(V, Q), Q(d)*  and therefore

Hn(Der *( V, Q; ")) is isomorphic to the dual of H*(V, Q(d)).
4    The Lie algebra structure


This section is devoted to the proof of Theorem 2 and its first applications.

For that, the following remark is essential:


Remark 7.   Let ( V, d) be a SullivanPmodel of a nilpotent space X. Recall

that d can be written as the sum d =   i 1di, with di(V )    iV . The linear

part d1 = Q(d) induces a differential on  V .  The differential d0 induced

by d on H*( V, d1) =  H*(V, d1) has no linear term and   H*(V, d1), d0

is the minimal model of X.  The quadratic part, d02is then a differential



                                   11

which can be identified as the Lie bracket on ss*(XQ ) [12, II.6.(16)].  More

precisely, given the natural isomorphism ss*(XQ ) ~= L* H*( V, d1), Q  and

the multilinear map


            <; , > : ^2H*( V, d1) x ss*(XQ ) x ss*(XQ ) ____- Q,


            <ff ^ fi; fl0, fl1> = fl1(ff)fl0(fi) + (-1)|fi||fl0|fl0(ff)fl1(fi),

it turns out that:


                   [fl0, fl1](ff) = (-1)p+q-1<d02ff; fl0, fl1>,


in which ff 2 H*( V, d1), fl0 2 ssp(XQ ), fl1 2 ssq(XQ ).

   In the same way, given the multilinear map


                  <; , > : ^jV x V *x . .x.V *____- Q,

                                          X
            <v1 . .v.j; fl0, . .,.flj> =      ffii1...ijfl1(vi1) . .f.lj(vij),

                                         i1,...,ij

where ffii1...ijis the expected sign induced by the Koszul convention, the high*
 *er

order Whitehead products on ss*(XQ ) can be identified with the i-th part of

d, via

            [fl1, . .,.flj](v) = (-1)p1+...+pj-1<div; fl1, . .,.flj> ,

being each fli of degree pi [1, Thm. 5.4] or [12, V.7(3)].


   Consider_now_the component F(X, Y ; f) of a given function space and

let ( SOE, d) be its Sullivan model defined in x2. We shall need a "quadratic"

analogue of Lemma 5. Given   2  V  and fi 2 B*, denote by {    fi}2 the

quadratic part of the image of [    fi] through the morphism:

                   ae-1                ________1            2
         A   B*=I ____~=- (V   B*) !  (V   B* )    (V   B*)  .


To effectively compute {    fi}2 use first the relations which generates I

to write [    fi] as an element of  (V   B*).  Then, cancel all elements of

negative degree and their derivatives, and replace any element of degree zero

by the corresponding scalar via OE. Finally, keep the quadratic part.


Lemma 8.   Let   = v1...vk 2  kV , fi 2 B* and ', _ 2 Der *( V, B; OE) of

strictly positive degrees. Then,
                                    X
<{  fi}2;  ',  _> = (-1)|fi|(|'|+|_|+| |)"ijfi OE(v1...^vi...^vj...vk)'(vi)_(vj*
 *) ,

                                     i6=j


                                   12

where ae0 = 0, aej = |v1| + . .+.|vj| for j   1, and
                     ae
                       (-1)|'|aei-1+|_|aej-1   ifi < j,
                "ij=       |'|ae   +|_|ae   +|'||_|
                       (-1)     i-1      j-1   ifi > j.


Proof.As in Lemma 5, to be clear in presenting our argument, we shall write

  instead of proper signs, and leave to the reader the straightforward task

that the equality above holds with the given signs.          P

   We proceed by induction on k. Let OE =Pv1v2, assume  fi =   r fi0r  fi00r
                                                  0||v |    0         00
and denote by   the sum of all terms of   r(-1)|fir (v21  fir)(v2   fir) in

which at least one of the two factors is of degree 0. Then
                                X
    <{v1v2   fi}2;  ',  _> =   <     (v1   fi0r)(v2   fi00r) -  ;  ',  _>.

                                 r

However, as ', _ are of positive degree, < ;  `g,  _> = 0 and the formula

above becomes:


        X
             <(v1   fi0r)(v2   fi00r);  ',  _> =

        Xr

      =      '(v1   fi0r) _(v2   fi00r)    _(v1   fi0r) '(v2   fi00r) =

        Xr

      =     fi0r('(v1))fi00r(_(v2))   fi0r(_(v1))fi00r('(v2)) =

         r    i                   j        i                   j

      =  FB B  '(v1)   _(v2)    fi    FB B  _(v1)   '(v2)    fi  =

      =  fi '(v1)_(v2)    fi _(v1)'(v2)


which is the expected expression for k = 2.

   Assume the lemma holds for k - 1 and let   = v1...vk. On one hand:


   X      i                                  j
        fi  OE(v1...^vi...^vj...vk)'(vi)_(vj)  =

    i6=j
   hX      i                                 X     i                           *
 *   j i
         fi  OE(v1...^vj...vk-1)'(vk)_(vj) +     fi  OE(v1...^vi...vk-1)'(vi)_(*
 *vk)   +

     j6=k                                    i6=k
       X        i                                  j
   +          fi  OE(v1...^vi...^vj...vk)'(vi)_(vj)  = (I) + (II).

       i6=j

       i,j6=k



                                   13

   On the other hand:



                              X
  <{v1...vk   fi}2;  ',  _> =     <{(v1...vk-1   fi0r)(vk   fi00r)}2;  ',  _>.

                               r


In this formula, whenever vk   fi00ris of degree 0, we can replace it by the

scalar OE(vk   fi00r) resulting:



            X
                   OE(vk   fi00r)<{v1...vk-1   fi0r)}2;  ',  _> +

          |vk fi00r|=0
           X
       +          <{v1 . .v.k-1  fi0r}(vk   fi00r);  ',  _> = (II0) + (I0)

         |vk fi00r|>0


   Applying induction we get:

           X       i                                   j
 (II0)  =       fi0rOE(v1...^vi...^vj...vk-1)'(vi)_(vj)  fi00r(OE(vk))

           i6=j,r
           X         i                                                        j
        =       FB B  OE(v1...^vi...^vj...vk-1)'(vi)_(vj)   OE(vk)   fi0r  fi00r

           i6=j,r
           X         i                                                  j
        =       FB B  OE(v1...^vi...^vj...vk-1)'(vi)_(vj)   OE(vk)    fi

            i6=j
           X       i                                               j
        =       FB  OE(v1...^vi...^vj...vk-1)'(vi)_(vj)OE(vk))   fi

            i6=j
             X        i                                          j
        =           fi  OE(v1...^vi...^vj...vk-1)OE(vk)'(vi)_(vj)  = (II).

              i6=j

              i,j6=k


On the other hand:

       X                                    X
(I0) =      '({v1...vk-1 fi0r}) _(vk fi00r)+      _({v1...vk-1 fi0r}) '(vk fi00*
 *r).

        r                                    r
                                   14

Applying Lemma 5 to this formula gives the following:

     X      i            j    i      j   X      i            j    i      j
   =     fi0r'(v1...vk-1)  fi00r_(vk)  +     fi0r_(v1...vk-1)  fi00r'(vk)

      r                                   r
     X        i                                j
   =     FB B  '(v1...vk-1)   _(vk)   fi0r  fi00r

      r       i                                j

         FB B  _(v1...vk-1)   '(vk)   fi0r  fi00r
           i                          j        i                          j
   =  FB B  '(v1...vk-1)   _(vk)    fi    FB B  _(v1...vk-1)   '(vk)    fi

                            j                         j
   =  fi '(v1 . .v.k-1)_(vk)    fi _(v1 . .v.k-1)'(vk)  .



   Finally, as ' and _ are OE-derivations, this last equation results in

X     i                              j  X      i                              j
    fi  OE(v1...^vi...vk-1)'(vi)_(vk)  +     fi  OE(v1...^vj...vk-1)_(vj)'(vk) *
 * = (I)

i6=k                                    j6=k

                                                                         __
and the proof is complete.                                              |__|


Proof of Theorem 2.  Let ', _ 2 Der ( V, B; OE) be homogeneous derivations

of positive degrees p and q respectively. In view of Theorem 1 and Remark

7, it is enough to show that, for any v   fi 2 SOE

                                         __
              [', _](v   fi) = (-1)p+q-1<d2(v   fi);  ',  _>

      __                                                              __
being d2, as always, the quadratic part of the differential in ( SOE, d).  But

this is trivial noting that ' and _ are of positive degree, and applying Lemma

8. Indeed:

                __                        p+q-1
      (-1)p+q-1<d2(v   fi);  ',  _> = (-1)     <{dv   fi}2;  ',  _>
                X
    = (-1)p+q-1    <{v1...vk   fi}2;  ',  _>
                X                  X        i                                  j
    = (-1)p+q-1    (-1)|fi|(|p+q+|v|+1)"ijfi  OE(v1...^vi...^vj...vk)'(vi)_(vj)

                                   i6=j
                           i          j
    = (-1)|fi|(p+q+|v|+1)fi  [', _](v)  =  [', _](v   fi).



To finish we show that the restriction to


      [ , ]: Der *( V, B+ ; OE)   Der *( V, B+ ; OE) -! Der *( V, B+ ; OE),



                                   15

also induces the Lie bracket in ssn(F*(X, Y ; f)Q ). For that note that, as the

fibration

                   F*(X, Y ; f) -! F(X, Y ; f) -!0!Y


has a section, the exact sequence on rational homotopy induces an extension

of Lie algebras


           0 ! ss*F*(X, Y ; f)Q ! ss*F(X, Y ; f)Q ! ss*YQ ! 0.


Hence, the Lie bracket on ss*F*(X, Y ; f)Q = H* Der ( V, B+ ; OE)  is the re-

striction of the one in ss*F(X, Y ; f)Q = H* Der ( V, B; OE) .


Remark 9.   At the sight of the proof above, which heavily relies on Remark

7, the fact that


        [ , ]: Der *( V, B; OE)   Der *( V, B; OE) -! Der *( V, B; OE)


commutes with differential automatically holds. This is far from trivial if one

uses only differential homological algebra tools.


   As a first and immediate application of Theorem 2 we describe the Lie

algebra structure on ss*F(X, Y ; *)Q and ss*F*(X, Y ; *)Q when considering the

constant map *: X ! Y , recovering in particular Vigu'e's result [14] stated

in the introduction.


Theorem 10.    ssn(F(X, Y ; *)Q ) (respec. ssn(F*(X, Y ; *)Q )) is isomorphic as

Lie algebra to H*(X; Q)   ss*(YQ ) (respec. H+ (X; Q)   ss*(YQ )).


Proof.In this case, OE: ( V, d) ! B annihilates V . In view of Theorem 2,

                        X           0                           00
[', _](v) = (-1)|'|+|_|-1  (-1)|_||vi|'(v0i)_(v00i) + (-1)|'|(|vi|+|_|)'(v00i)_*
 *(v0i),

                         i

           P
with d2v =   iv0iv00i. Via the_isomorphism  _of_Theorem 4, this is taken to

the Lie bracket induced by d2 on H*(V   B*, d1). However, this is precisely
                                                                        __
the V   H*(B) with the usual Lie bracket.                              |__|


   We may extend Lemma 5 to calculate in H* Der ( V, B; OE)  Whitehead

products of higher order.
                                   16

Definition 11.  Given '1, . .,.'j 2 Der *( V, B; OE), of strictly positive de-

grees p1, . .,.pj, define ['1, . .,.'j] 2 Der ( V, B; OE) by
      ['1, ..., 'j](v) =
                    X    X
      (-1)p1+...+pj-1        "i1...ijOE(v1 . .^.vi1...^vij...vk)'1(vi1)...'j(vi*
 *j) ,

                        i1,...,ij
           P
being dv =    v1 . .v.kand "i1...ijthe adequate generalization of "ij..

   Then, the exact analogue of the proof of Lemma 5 shows that given

  = v1...vk 2  kV and fi 2 B*,

   <{    fi}j;  '1, . .,. 'j> =
                      X
   (-1)|fi|(p1+...+pj+| |)"i1...ijfi OE(v1...^vi1...^vij...vk)'1(vi1)'2(vi2)...*
 *'j(vij) .

                     i1,...,ij

Again, {    fi}j is defined as the j-th part of the image of [    fi] through

the morphism:

                   ae-1                ________1            2
         A   B*=I ____~=- (V   B*) !  (V   B* )    (V   B*)  .


Thus, as in the proof of Theorem 2, we get the following which, in view

of Remark 7, describes j-order Whitehead products on ss*F(X, Y ; f)Q and

ss*F*(X, Y ; f)Q .
                                                    __
Theorem 12.     ['1, . .,.'j](v fi) = (-1)p1+...+pj-1<di(v fi);  '1, . .,. 'j>.



   From this, we immediately deduce Theorem 3.  For a given a space X,

recall that dlX (dl stands for differential length) is the least n, or infinite,

for which there is a non trivial whitehead product of order n on ss(XQ ). This

coincides with the least n for which dn, the n-th part of the differential of

the minimal model of X is non trivial. Another geometric description of this

invariant is given in [6] in terms of the Ganea spaces of X.


Proof of Theorem 3.  Assume cat0X = m. Then, by a deep result of Cornea

[3], X has a finite dimensional model B for which any product of length

greater than m of nonzero elements of B+ vanishes. Hence, for j > m and for

all v   fi, given '1, . .,.'j 2 Der ( V, B+ ; OE), ['1, . .,.'j](v_  fi) 2 B>m *
 * = 0.

However, as dlY > m, in view of Theorem 12, this implies that dj vanishes

for all j   2. This means that the differential on the minimal model vanishes

and the theorem follows.                                                .



                                   17

References


 [1]P. Andrews and M. Arkowitz, Sullivan's minimal models and higher

    order Whitehead products, Can. J. of Math., 30(8) (1978), 961-982.


 [2]E. H. Brown and R. H. Szczarba, On the rational homotopy type of

    function spaces, Trans. Amer. Math. Soc., 349 (1997), 4931-4951.


 [3]O. Cornea, Cone length and LS-category, Topology, 33 (1994), 95-111.


 [4]Y. F'elix, Rational category of the space of sections of a nilpotent bundle,

    Comment. Math. Helvetici, 65 (1990), 615-622.


 [5]Y. F'elix, S. Halperin and J.C. Thomas, Rational Homotopy Theory,

    G.T.M. 205, Springer, 2000.


 [6]Y. F'elix, and D. Tanr'e, When is the cohomology of a pointed mapping

    space a free algebra?, preprint, 2005.


 [7]A. Haefliger, Rational homotopy of the space of sections of a nilpotent

    bundle, Trans. Amer. Math. Soc., 273 (1982), 609-620.


 [8]Y. Kotani, Note on the rational cohomology of the function space of

    based maps, Homology, Homotopy and its Applications, 6 (2004), 341-

    350.


 [9]G. Lupton and S. Smith, Rationalized evaluation subgroups of a map

    and the rationalized G-sequence, preprint, 2004.


[10]G. Lupton and S. Smith, Rank of the fundamental groups of the com-

    ponents of a function space, preprint, 2004.


[11]A. Murillo and U. Buijs, Basic constructions in rational homotopy theory

    of function spaces, 2005, Submitted.


[12]D. Tanr'e, Homotopie Rationelle:  Mod`eles de Chen, Quillen, Sullivan,

    Lecture Notes in Math., 1025, Springer, 1983.


[13]R. Thom, L'homologie des espaces fonctionnels, Colloque de topologie

    alg'ebrique, Louvain, (1957), 299-39.


[14]M. Vigu'e-Poirrier, Sur l'homotopie rationnelle des espaces fonctionnels,

    Manuscripta Math., 56 (1986), 177-191.



                                   18

Departamento de 'Algebra, Geometr'ia y Topolog'ia

Universidad de M'alaga

Ap. 59, 29080 M'alaga

Spain

e-mail addresses: aniceto@agt.cie.uma.es, urtzi@agt.cie.uma.es

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