,
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
__ __
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 = (-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.
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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.
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
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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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