Basic constructions in rational homotopy
theory 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
Via the BousfieldGugenheim realization functor, and starting from
the Brown Szczarba model of a function space, we give a functorial
framework to describe basic objects and maps concerning the rational
homotopy type of function spaces and its path components.
1 Introduction
There are two choices to describe the equivalence between the homotopy cat
egories of rational nilpotent CWcomplexes of finite type and that of commu
tative differential graded algebras over the rationals. These choices depends
on how to realize a given algebra. On one hand, one may proceed as in [16],
considering a Sullivan model and then build its realization as a Postnikov
tower whose topological description mimics the algebraic properties of the
model. This is highly geometrical but quite unnatural. On the other hand,
one can take a general algebra and realizing it as a simplicial set via the
______________________________
*Partially supported by the Ministerio de Ciencia y Tecnolog'ia grant MTM200*
*406262
and by the Junta de Andaluc'ia grant FQM213.
2000 M.S.C.: 55P62.
Keywords: Function space, mapping space, Sullivan model, rational homotopy theo*
*ry.
1
BousfieldGugenheim realization functor [1]. This might be less transparent
but totally functorial.
In particular both options are available to the study of the rational homo
topy type of the function (or mapping) space F(X, Y ). The first approach
was first followed by Thom [18] and then by the fundamental work of Hae
fliger [8] where he supplied a model for the path component of sections,
homotopic to a fixed one, of a nilpotent bundle. In particular, a model of a
given component of a mapping space is produced. Later on, this approach
was further developed by other authors [4, 14, 15, 19]. However, due to the
non functorial description of the Haefliger model, it is not easy to use it to
describe further geometrical behavior of function spaces.
Under the second, and more functorial approach, it is essential the paper
of Brown and Szczarba [3] in which it is proved that the BousfieldGugenheim
realization of a particular commutative differential Zgraded algebra (CDGA
henceforth), which already appears in [8], is homotopy equivalent to the (non
connected!) function space F(X, Y ). It is also shown how to obtain from
this algebra a quotient CDGA whose realization is a given path component
of the function space, and therefore, the Haefliger model of this component
is retrieved.
In this paper we continue with the study under this natural approach, and
present models for some of the basic objects and maps concerning function
spaces. By a model of a non connected object (or a map between them) we
mean a Zgraded CDGA (or a morphism) whose simplicial realization has the
homotopy type as the singular simplicial approximation of the chosen object.
We also present an explicit way of restricting algebraically to components
obtaining Sullivan models of nilpotent spaces and maps relating functions
spaces, retrieving in particular the Haefliger model for its components.
We now state our main results: For any function space F(X, Y ) we shall
always assume X and Y CWcomplexes with X finite, so that F(X, Y ) is
itself a CWcomplex [13], and with both X and Y nilpotent of finite type over
Q, so that the components of F(X, Y ) are also nilpotent and of finite Qtype
[10, Thm.2.5]. Given f :X ! Y , denote by F(X, Y ; f) the component of
F(X, Y ) containing f. We first give an explicit model of the evaluation map
which includes in particular the original result of Haefliger [8, Thm.3.2] and
[11, Thm.1.1].
Theorem 1. Let ( (V B*), ed) be the BrownSzczarba model for F(X, Y )
2
(see x2). Then,
X
! :( V, d) ! ( (V B*), ed) B, !(v) = (1)ffbi(v fii) bi,
i
where {bi} is a basis for B, {fii} its dual and ff(n) is the integer part of
(n + 1)=2, is a model of the evaluation map
! :F(X, Y ) x Y ! Y, !(f, x) = f(x).
In particular, the map
!0: ( V, d) ! ( (V B*), ed), !0(v) = v 1*,
is a model of the evaluation of the base point !0: F(X, Y ) ! Y , !0(f) =
f(x0).
Moreover, given i :( V, d) ____ ( W, d) a Sullivan model for f :X ! Y ,
there is an explicit morphism , :( (V C*), ed) ____ ( (W C*), ed) (see x3)
between the models of F(Z, Y ) and F(Z, X) satisfying:
Theorem 2. The commutative square
, 1C
( (W C*), ed) C oe___________( (V C*), ed) C
6 6
!  !
 
 
( W, d) oe_______________________( V, d)
i
is a model of
(f)* x 1Z
F(Z, X) x Z _____________F(Z, Y ) x Z
 
!  !
 
? ?
X _______________________Y.
f
3
Next, we show how to obtain from these results the corresponding Sullivan
models when considering a fixed component of the function space. Indeed,
given ( (V B*), ed) the model of F(X, Y ) and a map f :X ! Y , there
is a Sullivan model (in fact this_is the Haefliger model) of the component
F(X, Y ; f) of the form ( Sf, d) in which Sf2 = (V B*) 2 and S1fis a
quotient of (V B*)1 (see x4). Then we prove:
Theorem 3. The CDGA morphism
__
!0: ( V, d) ____ ( S, d),
given by !0(v) = v 1* if v 2 V 2, or its projection over S1fif v 2 V 1, is
a Sullivan model of the evaluation at the base point !0: F(X, Y_;_f) ____ Y .
Moreover, with the notation of Theorem 2, , induces a morphism , for which
the square
__
__ , __
( Sg, d) oe_______________ ( Sfg, 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)
 
!0  !0
 
? ?
X _______________________ Y.
f
As in this result, !0 is a fibration with fibre F*(X, Y ; f), the space of
pointed maps, we easily obtain the following consequence, a form of which
already appeared in [4], [11] or [19].
__
Corollary 4. The CDGA ( (Sf=V ), d) is a Sullivan model of F*(X, Y ; f).
Finally, we translate the constructions above to rational homotopy groups
by taking the homology of the indecomposables of the given sullivan models
and maps.
4
2 Notation, basic facts and tools
All spaces considered henceforth shall be of the homotopy type of connected
CWcomplexes. We will heavily rely on known results from homotopy theory
of simplicial sets and from rational homotopy theory. For that [7] or [12] and
[6] are respectively standard and excellent references. Here, mainly to set
the framework in which we work and to fix notation, we recall some basics:
Denote by
S*
Top _____oe____SimpSet
 . 
the singular and realization adjoint functors relating the categories of sim
plicial sets and topological spaces. These induce equivalences between the
(pointed if we wish) homotopy category of CWcomplexes and the homotopy
category of Kan complexes [13]. For each space X and each simplicial set
K, We shall denote by _(K): K  '! S*K and OE(X): S*X '! X the
corresponding weak equivalences.
On the other hand, if CDGA is the category of cochain Zgraded com
mutative differential graded algebras, there are adjoint functors [1]
APL
SimpSet _____oe____CDGA,
<.>
where: APL (X) is the CDGA of piecewiselinear polynomial forms on X
with rational coefficients and is the simplicial realization of the CDGA
A: For any q 0, q = hom (A, (APL )q), i.e., the set of morphisms from
A to (x1, . .,.xq, dx1, . .,.dxq). The face and degeneracies operators are
induced by their counterparts in the simplicial algebra (APL )* [1, 6].
the composition of the above pairs of adjoint functors produce equiva
lences when restricted to the homotopy category of rational nilpotent space
of finite type over Q and the homotopy category of connected CDGA's over
Q of the same homotopy type of Sullivan algebras of finite type [1, 6]. Abus
ing of notation we shall write APL (X) = APL (S*X) and A =  for
any space X and any CDGA A. Given a non necessarily connected space
Z, a CDGA A is a model of Z if S*(Z) and are simplicial sets of the
same homotopy type. As we are mainly concerned in this paper on map
ping spaces, let us describe these objects whenever we take simplicial sets
or CDGA's. Given K, L simplicial sets, F(K, L) is the simplicial set defined
5
by F(K, L)q = hom (K x 4 [q], L), where 4 [q]2SimpSet is the standard
nsimplex: For each n 0, [n] = {0, . .,.n} and (4 [n])q is the set of non
decreasing maps [q] ! [n]. Call 4q to 1[q]2 4 [q]q. Faces and degeneracies
operators on F(K, L) are defined by @if = f O (1K x ffii) y sif = f O (1K x oei*
*).
Whenever L is Kan, F(K, L) is also Kan and it has the same homotopy type
of S*F(K, L) via the equivalence
ff :F(K, L) '! S*F(K, L),
in which ff(f): 4qF(K, L) is given by the exponential law applied to the
composition
' ~= f
K x 4q ____ K x 4[q] ____ K x 4[q] ____ L.
The evaluation map ! :F(K, L) x K ! L is defined by !(f, oe) =
f(oe, 4q), where 4q = (0, 1, . .,.q) 2 4 [q]qis the fundamental qsimplex.
This map can be identified with S*! :S*(F(K, L)xK) ~=S*F(K, L)x
S*K ! S*L via the following result obtained by direct calculation:
Proposition 5. The following diagram commutes:
ff x _(K)
S*F(K, L) x S*K oe___________F(K, L) x K
 ' 
 
S*!  !
 
? _(L) ?
S*L oe________________________L.
'
On the other hand given CDGA's A and B define F(A, B) 2SimpSet
by F(A, B)q = hom (A, (APL )q B) with faces and degeneracies given by
@if = (@i 1B ) O f, sjf = (sj 1B ) O f. If OE: B ! C is a CDGA morphism,
we denote by
OE*: F(A, B) ! F(A, C)
the simplicial map given which assigns to each morphism _ :A ! (APL )q B
the composition OE*(_) = (1(APL)q OE) O _. If A, B, C are connected CDGA's
and OE is a quasiisomorphism, then OE* is a weak equivalence.
Next we recall the BrownSzczarba model of the function space F(X, Y ),
with X as always a finite complex and both X and Y nilpotent of finite
6
type (over Q). Consider A = ( V, d) '! APL (Y ) a Sullivan model (non
necessarily minimal!) of Y and B '! APL (X) a quasiisomorphism with B
a connected CDGA of finite type. Let B* = hom (B, Q) be the differential
graded coalgebra dual of B, and consider the 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)a2fij(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 [3, Thm.1.2], and therefore, considering
on (V B*) the differential ed= ae1dae, ae is also an isomorphism of CDGA's.
Then, ( (V B*), ed) is a model of F(X, YQ ) [3, Thm.1.3]. In other words,
S*F(X, Y ) and <( (V B*), ed)> are homotopy equivalent.
In order to explicitly determine edon v fi 2 V B*, calculate (dv)
fi + (1)vv dfi and then use the relations which generate the ideal I to
express (dv) fi as an element of (V B*).
3 Compatible models for the evaluation map
In this section we properly state and prove Theorems 1 and 2. For that, fix
as before ': ( V, d) !' APL (Y ) a Sullivan model of Y (i.e., a KScomplex
'
non necessarily minimal), and fi :B _____ APL (X), a quasiisomorphism in
which B is finite dimensional and satisfies and additional condition: the
inclusion of the base point {x0} ,! X determines a canonical augmentation
": APL (X) ! APL (x0) ~= Q. Thus we may, and will, choose B connected
and fi preserving augmentations, i.e., " O fi(1) = 1. Finally, fix {bi} (respec.
{fii}) a basis for B (respec. the dual basis of B*) and denote by ff(n) the
integer part of (n + 1)=2. Then we have:
Theorem 6. the CDGA morphism
X
! :( V, d) ! ( (V B*), ed) B, !(v) = (1)ffbi(v fii) bi,
i
7
is a model of the evaluation map
! :F(XQ , YQ ) x XQ ! YQ , !(f, x) = f(x).
That is to say, there is a commutative diagram like the following in which
the vertical arrows are equivalences:
!
F(XQ , YQ ) x XQ ____________________YQ. Top
...
.S*?...
S*!
S*F(XQ , YQ ) x S*XQ _______________S*(YQ )
 
'  ' SimpSet
? ?
<( (V B*), ed) B>______________<( , V, d)>
.6...
.<.>...
!
( (V B*), ed) Boe_______________( V, d) CDGA
As the inclusion i: F(X, Y ) ,! F(X, Y )xX on the base point x0 is clearly
modeled by ( (V B*), ed) B 1"!( (V B*), ed), and since !0 = ! O i, we
immediately have:
Corollary 7. The CDGA morphism
!0: ( V, d) ____ ( (V B*), ed), !0(v) = v 1*,
is a model of the evaluation map at the base point !0: F(X, YQ ) _____ YQ ,
!0(g) = g(x0).
Note that in Theorem 6 we consider F(XQ , YQ ) while in Corollary 7 the
evaluation map is defined over F(X, YQ ): although there is a weak equiva
lence [15, Thm.2.3] F(XQ , YQ ) ' F(X, YQ ), in the first result we evaluate at
any point of XQ while in the second we simple do it at the base point of X.
Proof.Start by choosing simplicial sets K = S*(XQ ), L = S*(YQ ). As Y is
nilpotent, the natural map induced by the adjunction j :L '! ,
j(oe)( ) = oe, is an equivalence which may be composed with the map
8
<'>: '! to produce an equivalence h: L '! . Apply just
definitions to see that the following commutes:
h* x 1K
F(K, L) x K ____________F(K, ) x K
'
 
!  ! (1)
? ' ?
L _______________________.
h
Next, we use the homotopy equivalence fl :F(A, APL (K)) '! F(K, )
given in [2, Thm.20] which we now describe: Consider the morphism
APL (p1) APL (p2): APL (K) APL (4 [q]) ____ APL (K x 4 [q]),
p1 p2
induced by the projections K oe___ Kx4 [q]____ 4 [q], and the isomorphism
of [6, Prop.10.4] ` :(APL )q ____ APL (4 [q]) which assigns to each 2 (APL )q
the only element `( ) 2 APL (4 [q]) satisfying `( )(4q) = . Then, given
_ :A ____ (APL )q APL (K) an element of F(A, APL (K))q,
fl(_) = O j :K x 4 [q]_____ .
As before, j :K x 4 [q]'! is the natural equivalence
induced by adjunction. On the other hand, define a simplicial map
__!:F(A, A _____
PL (K)) x
which sends each _ :A ____ (APL )q APL (K) and OE: APL (K) ! (APL )q to
the composition
__!(_, OE): A _____(A ____1.OE
PL )q APL (K) (APL )q.
It is then straightforward to check that the following commutes:
fl x j1
F(K, ) x K oe___________F(A, APL (K)) x
 ' 
!  __! (2)
 
? ?
=========================== .
9
For the simplicial map __!:F(A, B) x ! , defined as before by
__!(_, OE) : A _____(A ____1.OE
PL )q B (APL )q,
it is also easy to see the commutativity of:
fi* x
F(A, APL (K)) x oe___________F(A, B) x
 ' 
__! __ 
 !  (3)
 
? ?
========================== .
Finally, consider the isomorphism of CDGA's
~=
ae: ( (V B*), ed) ____ ( (A B*)=I, d),
recalled in x2, and the simplicial isomorphism [3, Thm.3.1]
~=
: < (A B*)=I> ! F(A, B),
defined as follows: for each j 2 < (A B*)=I>q and each v 2 V ,
X
(j)(v) = (1)ffbij [v fii] bi.
i
On the other hand observe that
< (A B*)=I B> ~=< (A B*)=I> x .
Then, under this identification, we may show that the following commutes:
x 1
< (A B*)=I> x ____________F(A, B) x
 
<(ae 1B ) O !> __! (4)
 
? ?
======================== .
Indeed, given each j 2 < (A B*)=I>q, OE 2 and v 2 V :
X
(__!O x 1)(j, OE)(v) = __!( (j), OE)(v)=(1)ffbij [v fii]. OE(bi)
i
= <(ae 1B ) O !>(j, OE)(v).
10
To finish, use diagrams (1), (2), (3) and (4), together with proposition 5,
to obtain the following diagram which completes the proof:
!
F(XQ , YQ ) x XQ ________________________YQ. Top
...
.S*?..
S*(!)
S*F(XQ , YQ ) x S*XQ ____________S*(YQ )
6 6
' ' 
 ! 
F(K, L) x K __________________________L
 
h* x 1K ' ' h
? ! ?
F(K, ) x K ________________________w
6 ww
fl x j1' ww
 __!
F(A, APL (K)) x ___________________w SimpSet
6 ww
fi* x ' ww
 __!
F(A, B) x _________________________w
6 ww
x 1 ~= ww
 <(ae 1B ) O !>
< (A B*)=I> x _____________________w
 ww
x 1~= ww
?
< (V B*)> x ______________________<.V >
.6..
..<.>..
...
.
(V B*) B oe_______________________ V AGDC
!
__
__
Next, fix a map f :X ! Y between nilpotent complexes of finite type over
'
Q and let Z be a finite nilpotent complex. Let A = ( W, d) _____'APL (X)
_
and B = ( V, d) _____'APL (Y ) be Sullivan models (again non necessar
11
ily minimal!) of X and Y respectively, let C = (C, ffi) _____'APL (Z) be a
quasiisomorphism with C finite dimensional connected of finite type, and
let i :( V, d) ____ ( W, d) be a Sullivan model for f. Define
, :( (V C*), ed) ____ ( (W C*), ed), ,(v c) = ae1[i(v) c],
being ( (V C*), ed) and ( (W C*), ed) the models of F(Z, YQ ) and
~=
F(Z, XQ ) respectively, and ae: ( (W C*), ed) _____ ( (A C*), d)=I the
CDGA isomorphism described in x2. In other words, to compute effec
tively ,(v c) use the relations which define I to express i(v)P c as an
element of (V C*). For instance, if i(v) = w1w2 and 4c = ic0i c00j,
P 0
,(v c) = i(1)w2ci(w1 c0i)(w2 c00i).
Then, we prove this more explicit version of Theorem 2:
Theorem 8. The commutative square
, 1C
( (W C*) C, ed)oe___________( (V C*) C, ed)
6 6
!  !
 
 
( W, d) oe______________________ ( V, d)
i
is a model of
(fQ )* x 1
F(ZQ , XQ ) x ZQ____________F(ZQ , YQ ) x ZQ
 
!  !
 
? ?
XQ ________________________YQ ,
fQ
i.e., there exist a commutative cube of simplicial sets as the following in whi*
*ch
12
the vertical arrows are homotopy equivalences:
S*((fQ )* x 1)
S*(F(ZQ , XQ ) x ZQ )__________________________ S*(F(ZQ , YQ ) x ZQ )
 Z  Z
Z  Z
 Z '  Z
 Z  Z
 S*(!) Z S*(!) Z
 Z" S*(fQ )  Z"
'  S*(XQ ) ___________________________________S*(YQ )
   
   
   
 '   
   
?  ? 
< (W C*) C> _____________________________<(V C*) C> '
Z  <, 1C > Z 
Z  Z 
Z  Z 
Z  Z 
Z  Z 
Z" ? Z" ?
< W, d> __________________________________< V, d>.
As in Corollary 7, evaluating at the base point we immediately obtain:
Corollary 9. The commutative square
,
( (W C*), ed)oe___________( (V C*), ed)
6 6
!0  !0
 
 
( W, d) oe_________________ ( V, d)
i
is a model of
(fQ )*
F(Z, XQ ) ________________F(Z, YQ )
 
!0  !0
 
? ?
XQ ______________________YQ .
fQ
Proof.Fix N = S*(ZQ ), K = S*(XQ ), L = S*(YQ ) and ~ = S*(fQ ): K ____ L.
Then we have:
13
Lemma 10. The following commutes:
S*~* x 1
S*F(N, K) x S*N ________________________ S*F(N, L) x S*N
 "Z  "Z
Z  Z
 Z ' S (!)  Z '
 Z *  Z
 ff x _(N) Z ff x _(N) Z
 Z ~* x 1  Z
S*(!)  F(N, K) x N ____________________________F(N, L) x N
   
   
   
 !   
   
?  S*(~) ? 
S*(K) _________________ ____________________S*(L) !
"Z  "Z 
Z  Z 
Z  Z 
Z  Z 
' Z  ' Z 
Z ? ~ Z ?
K _______________________________________L.
Proof.the lateral faces commute by Proposition 5, while the back and bot
tom faces are trivially commutative. For the front face observe that given
g :N x 4 [q]______ K in F(N, K)q, (~ O !)(g, oe) = ~(g(oe, 4q)) = !*
* O
(~* x 1)(g, oe). Finally, to check the commutativity of the top face, cho*
*ose
as before g 2 F(N, K)q and note that ff(g) 2 SqF(N, K) is the map
4q ____ F(N, K) associated by the exponential law to
' ~= g
o :N x 4q ____ N x 4 [q] ____ N x 4 [q] ____ K.
Taking into account that ~*(g) = ~ O g = ~ O g, we observe that
ff(~*(g)) 2 SqF(N, L) is again associated by the exponential law to
1x' ~j ~Og
o O ~: N x 4q ____ N x 4 [q] ____ N x 4 [q] _____ L.
*
* __
Then, (S*(~*) O ff)(g) = ~* O ff(g) = ff(~*(g)) = (ff O ~*)(g). *
* __
14
Lemma 11. The following commutes:
~* x 1
F(N, K) x N ___________________________F(N, L) x N
 Z  Z
 Z  Z
 Z (hK )* x 1 !  Z (hL)* x 1
 Z  Z
 ' Z ' Z
 Z" x 1  Z"
!  F(N, ) x N _____________________*F(N, ) x N
   
   
   
 !   
   
?  ~ ? 
K _____________________________________L !
Z  Z 
Z Z
Z hK  Z hL 
Z  Z 
' Z  ' Z 
Z" ? Z" ?
________________________________.
Proof.Side faces are square of type (1); commutativity of the back face has
been checked in Lemma above while that of the front and bottom face are
trivial. For the top face, given g 2 F(N, K)q, (hL)* ~*(g) = hL O~Og, while
* (hK )*(g) = O hK O g. But these are equal in view of the following
commutative diagram in which the composition of the vertical arrows are
precisely hK and hL:
~
K _________________________L
 
 
'  '
 
? ?
_____________
 
<'>  <_>
 
? ?
_______________________.
__
__
15
Lemma 12. The following commutes:
* x 1
F(N, ) x N _____________________________________F(N, ) x N
 "Z  "Z
Z 1  Z 1
 Z flA x j !  Z flB x j
 Z B  Z
 ' Z ' Z
 Z i* x 1  Z
!A  F(A, APL (N)) x _____________________F(B, APL (N*
*)) x
   *
* 
   *
* 
 __   *
* 
 !A   *
* 
   *
* 
?  ? *
* __
_______________________________________________ *
* !B
l  l *
* 
l  l *
* 
l  l *
* 
l  l *
* 
l ? l *
* ?
___________________________________________ *
*.
Proof.The lateral squares are as in (2). For the front face, given *
*g 2
F(A, APL (N))q, both (__!BO i* x 1)(g, OE) and ( O __!A)(g, OE) a*
*re the same
composition
i g 1.OE
B ____ A ____ (APL )q APL (N) ____ (APL )q.
For the top face recall that flA :F(A, APL (N)) ! F(N, ) assigns *
*to each
g :A ! APL (N) (APL )q the simplicial map fl(g) = O j in which j :N x 4 [q]_____ APL (N x 4 [q])> is induc*
*ed by
adjunction. Taking into account that = = O =*
* *(g),
it is immediate to see that both (* O flA )(g) and (flB O i*)(g) *
*correspond to
fl(g) *
* __
the same map N x 4 [q]____ ____ . *
* __
16
Lemma 13. The following commutes:
i* x 1
F(A, APL (N)) x _________________________F(B, APL (N)) x
 kQQ __ kQ
 Q Q*x < > !B  Q Q*
* * x < >
__  ' Q i* x 1  ' *
* Q Q
! A F(A, C) x ________________________________*
*F(B, C) x
   *
* 
 __   *
* 
 !A   *
* 
?  ? *
* __
_______________________________________________ *
* !B
Q Q  Q *
* 
Q Q  Q Q*
* 
Q ? *
* Q Q ?
_______________________________________*
*____.
Proof.Lateral faces are squares like (3). The front face has been ch*
*ecked in
*
* __
the past Lemma and the rest of the faces are trivially commutative. *
* __
Lemma 14. The following commutes:
i* x 1
F(A, C) x _______________________________________F(B, C) x
 kQQ __ kQ
 ~Q QAx 1 !  Q~Q B x 1
__  = Q <[ (i 1)]> x 1  = Q Q
!  < A C*=I> x ______________________________< B C*=*
*I> x
   *
* 
   *
* 
   *
* 
?  ? *
* 
_________________________________________________ *
*
Q Q  Q *
* 
Q Q  Q Q *
* 
Q ? Q Q *
* ?
_____________________________________________<*
*B>.
Proof.First, observe that the morphism (i 1): (B C*) ! (A
C*) sends IB into IA and therefore it induces [ (i 1)]: (B C*)=I*
*B !
(A C*)=IA . Now, the back face has been checked checked in the p*
*ast
Lemma while lateral faces are squares like (4). For the top face obs*
*erve that
given g 2 < (A C*)=IA >qand b 2 B, i*( A (g))(b) = ( A (g) O i)(b)*
* =
P ff(ci) *
i(1) g([ib ci]) ci.
17
On the other hand,
X
B (<[ (i 1)]>(g))(b) = (1)ff(ci)<[ (i 1)]>(g)([b c*i]) *
* ci
Xi
= (1)ff(ci)(g O [ (i 1)])([b c*i])*
* ci
Xi
= (1)ff(ci)g([ib c*i]) ci.
i
Finally, the square
[ (i 1)] 1C
(A C*)=IA C oe______________ (B C*)=IB C
6 6
ae 1C O ! ae 1C O !
 
 i 
A oe______________________________B,
*
* __
is trivially commutative, so is the front face. *
* __
We shall also need an additional cube whose commutativity is trivial:
Lemma 15. The following commutes:
<[ (i 1)] 1C >
< A C*=I C> _______________________< B C*=I C>
 Q Q Q
 Q Q  Q Q
 ' Qs <, 1C >  ' Q Qs
 < (W C*) C> __________________________< (V C**
*) C>
   
   
   
?  ? 
___________________ ________________________ 
Q Q  Q 
Q Q  Q Q 
Q ? Q Q ?
< W > _____________________________________< V *
*>.
To finish the proof of Theorem 8 join the cubes of past lemmas by the
*
* __
back and front face to obtain the required commutative diagram. *
*__
18
4 The restriction to components
Here we restrict the results of past sections to the components of function
spaces to get, in particular, Theorem 3 of the introduction. For this we need
some algebraic tools: let ( W, d) be a CDGA in which W is Zgraded, and
let u: W _____ Q be an augmentation. In ( W, d) we consider as in [3]
the differential ideal Ku generated by A1 [ A2 [ A3, where A1 = ( W )<0,
A2 = d( W )0 and A3 = {ff  u(ff) : ff 2 ( W )0}.
Lemma 16. The ideal Ku coincides with Ku 0generated by A1 0[ A20[ A30,
where A1 0= W <0, A2 0= dW 0and A3 0= {w  u(w) : w 2 W 0}.
Proof.The inclusion Ku 0 Ku is trivial, as it is A1 Ku 0.
Let = ffu(ff) 2 A3 and write ff = a+b, a 2 + (W <0). W , b 2 W 0.
Then, ff  u(ff) = a + b  u(a)  u(b) = a + b  u(b). As a 2 A1 Ku 0, it
remains to see that bu(b) 2 Ku 0. Assume b 2 nW 0and argue by induction
on n. For n = 1, trivially b  u(b) 2 A3 0 Ku 0. Let b = b1 . .b.n, bi 2 W 0.
Then, b  u(b) = b1 (b2 . .b.n)  u(b2 . .b.n) + u(b2 . .b.n)(b1  u(b1) 2 Ku 0
by induction hypothesis.
Finally, let = dff, ff 2 ( W )0, be a generator of A2 and write ff =
a + b where a 2 ( + W <1) . W , b 2P (W 0) and c 2 (W 0) . W 1. W 1.
Obviously da, db 2 KuP0. Write c = c1c2, 2 (W 0), c1 = 1 y
c2 = 1. Thus, dc = d c1c2 + dc1c2  c1dc2. While the first and third
set of summands are trivially in Ku 0, u(dc1) = du(c1) = 0 and therefore
__
OEdc1c2 = OE(dc1  u(dc1))c2 is also in Ku 0. _*
*_
Next, we see that ( W, d)=Ku is itself a free commutative graded algebra.
Let fKu be the ideal of ( W, d) generated by A1 [ A3 and observe that the
~=
projection ae: W 1! ( W=Kfu)1 is a vector space isomorphism. Consider the
linear map
d 1 1 ae1 1
@ :W 0____ ( W ) ___( W=Kfu) ____ W
___1 ___1
and call W a complement of the image of this map, W 1= @W 0 W .
In which follows, given = ff . , ff 2 ( + W 0) and 2 W , we denote
by =u the element u(ff) . Hence, if for w 2 W 0, dw = 0 + 1 + 2, with
0 2 ( + W <0).( W ), 1 2 ( + W 0).W 1, 2 2 W 1, then @(w) = 1=u+ 2.
__ ___1 __
Proposition 17. For a certain d, ( W=Ku, d) ~= (W W 2, d).
19
Proof.It is_easy to see that the surjective morphism of graded algebras
1 2
': W ! W W , given by
8
< 0 ifw< 0 or w 2 @W 0 ,
'(w) = u(w) ifw = 0,
: w otherwise,
has_Ku as kernel so it induces the required isomorphism. To finish, define
__
d= __'O d O __'1. __
__ ___1
Remark 18. (1) To effectively compute d, choose w 2 (W W 2) and
write dw = 0+ 1+ 2+ 3, in which 0 2 + W <0. W , 1 2 + (@W 0) .
___1 ___1 __
W 0, 2 2 ( + W 0) . ( W W 2) and 3 2 W W 2. Then, dw =
__'d__'1w = '(
2 + 3) = 2=u + 3.
(2) Note that if we have in W a basis wi for which dwi 2 W, i.e., the CDGA morphism u: ( (V B*), ed) ! Q, which cor
responds to hf through the equivalence S*F(X, YQ ) '! < (V B*)>.
We first prove a generalization of [3, Thm.6.1]:
20
Proposition 19. The projection ( (V B*), ed) ! ( (V B*), ed)=Ku in
duces a homotopy equivalence
'
< (V B*)=Ku> _____ < (V B*)>u
which makes the following commutative:
< (V B*)>u ____< (V w B*)>
6 ww
' ww
 w
< (V B*)=Ku> ___< (V B*)>.
Proof.We shall need the following version of [9, Thm. 2.2] for ( (V B*), "d):
Write B* = A ffiA C in which C ~=H(B*) and choose a basis {aj, bj, ck}
with ffiaj = bj and ffick = 0. Also, as V is a Sullivan model choose {vi} a
basis for V satisfying dvi 2 (V ~= <( W, d)=Kp (U dU)=Kq>
~= <( W, d)=Kp> x < (U dU)=Kq>
' <( W, d)=Kp>.
Finally, in [3, Thm.6.1] is proved that the projection W ! W=Kp induces
an equivalence < W=Kp> '! < W >p. Therefore, as < (V B*)>u ' < W >p
we get an equivalence < (V B*)=Ku> '! < (V B*)>u which satisfies the
__
required property. __
Then, we have:
Theorem 20. The projection ( (V B*), ed) ! ( (V B*), ed)=Ku is a
model of i : F(X, Y ; f) ____ F(X, Y ).
Proof.Indeed, we have the following diagram
i
F(X, YQ ; h O f)____.F(X, YQ ) Top
..
.S*?..
S*i
S*F(X, YQ ; h O f) ___S*F(X,wYQ )
6 ww
'  ww

S*F(X, YQ )hf _____S*F(X, YQ )
'  ' SimpSet
? ?
< (V B*)>u _____< (Vw B*)>
6 ww
' ww

< (V B*)=Ku> ____< (V B*)>
.6...
<.>....
...
(V B*)=Ku oe___ (V B*) AGDC
in which the top and middle square are trivially commutative and the bottom
__
is given in Proposition 19. __
22
The next results prove Theorem 3 and Corollary 4.
Theorem 21. The CDGA morphism
!0: ( V, d) ____ ( (V B*), ed)=Ku, !(v) = [v 1*],
is a model of !0: F(X, Y ; f) ____ Y.
Proof.Write !0 as the composition
i !0
F(X, Y ; f) ____ F(X, Y ) ____ Y
__
and apply Theorems 6 y 20. __
In particular, via Proposition 17, we have:
Corollary 22. The CDGA morphism
__ _______1 2 __
!0: ( V, d) ____ ( Su, 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.
Observe that, while !0(v) could vanish if v = 1, when ( V, d) is 1
connected,
__ _______1 2 __
!0: ( V, d) ____ ( Su, d) = ( V B * (V B*) , d),
is a KSextension or a relative Sullivan algebra. The fibre is of the form
_______1 2 * __ __
( (V B * (V B*) )=(V 1 ), d) = ( (Su=V ), d).
Hence, as the fibre of !0: F(X, Y ; f) ! Y is F*(X, Y ; f), the path compo
nent of f of the space of pointed maps, we immediately obtain:
__
Corollary 23. For a 1connected space Y , ( (Su=V ), d) is a Sullivan model
of F*(X, Y ; f).
Finally, let X, Y, Z and f :X ! Y be as in Theorem 8. Let ug: (W
C*) ! Q and ufOg: (V C*) ! Q be the 0simplices of < (W C*)> and
< (V C*)> corresponding to hOg :Z ! XQ and hOf Og = fQ OhOg :Z ! YQ
respectively.
23
Theorem 24. The square
__ ~, __
( Sug, d)oe_______________( Sufg, 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)
 
!0  !0
? ?
X _______________________ Y,
f
~,
Proof.First observe that the (W C*)=Kug oe__ (V C*)=KufOgis well
defined, i.e., ,(KufOg) Kug. For that note that ufOgis the composition ugO,.
If A1 0, A2 0y A3 0is the set of generators of KufOggiven in Lemma 16, then
,(A1 0) Kug trivially. Let ed(w c), w c 2 (W C*)0, an element of A2 0.
Then, ,(de(w c)) = ed,(w c) = edae1 [iw c]. Write ae1 [iw c]= a + b,
a 2 + (V C*)<0. (V C*), b 2 (V C*)0. Hence, ed(a), ed(b) 2 Kug.
Finally consider w c  ufOg(w c), w c 2 (W C*)0, a generator of A30.
Then,
,(w c  ufOg(w c)) = ,(w c)  ufOg(w c) = ,(w c)  ug(,(w c)),
which again, by Lemma 16, lives in Kug. To finish the proof we have to show
the existence of a commutative cube in SimpSet as the following, in which
the vertical arrows are homotopy equivalences:
24
S*(fQ )*
S*F(Z, XQ; h O g)____________________S*F(Z, YQ ; fQ O h O g)
 ZZ  Z
 Z Z '  Z Z
 S*! Z Z"  S*! ZZ Z"
 S*(fQ ) 
'  S*(XQ ) ________________________________S*(YQ )
   
   
   
 '   
   
?  ? 
< (W C*)=Kug> ____________________< (V C*)=KufOg> '
Z  <~,> Z 
Z Z  Z 
! ZZ  Z!Z 
Z" ? Z Z" ?
< W > _________________________________< V >.
But this is immediate: Lateral faces are obtained by Theorem 21. The
commutativity of the back face is an exercise and the rest are trivially com
__
mutative. __
Finally, we describe the behavior at the homotopy group_level of the
distinguished maps studied in past sections. Let ( S, d) ~=( (V B*)=Ku, d)
be the Sullivan model of the component F(X, Y ; f) given in Corollary 22.
Recall that u:__(V B*)_!_Q_is_the CDGA morphism_corresponding to f
and that ( S, d) = ( (V B* )1 (V B*) 2, d). Now, by classical facts on
rational homotopy theory of nilpotent spaces, we have:
Theorem 25._ (1) ss*F(X,_Y ; f)Q is naturally_isomorphic to the dual of
H* S, Q(d) ~= H*(V B* )1 (V B*) 2, Q(d), being S ~= Q( S) =
S=( + S . + S) the space of indecomposables. Moreover, the morphism
ss*(!0): ss*F(X, Y ; f)Q ! YQ is dual of
* * ________1 2 __
H* Q(!0) :H V, Q(d) ! H V B* ) (V B*) , Q(d) ,
H* Q(!0) (v) = [v 1*].
(3) under the conditions and with the notation of Theorems 8 or 24, the
25
morphism induced in homotopy groups by
(f*)Q
F(Z, X; g)Q _____________F(Z, Y ; f O g)Q
 
!0  !0
? ?
XQ _______________________YQ ,
fQ
is dual of
__ H*(Q(,)) * __
H*(Sug, Q(d)) oe__________H (Sufg, Q(d))
6 6
H* Q(!0)  H*Q(!0)
 * 
H* W, Q(d) oe____________H V, Q(d) .
H*(Q(i))
Also, taking homology of indecomposables in Corollary 23 we obtain:
Corollary 26._ ss*F*(X, Y ; f)Q is naturally isomorphic to the dual of
H* S=V, Q(d) .
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27
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Departamento de 'Algebra, Geometr'ia y Topolog'ia
Universidad de M'alaga
Ap. 59, 29080 M'alaga
Spain
email addresses: aniceto@agt.cie.uma.es, urtzi@agt.cie.uma.es
28