ON THE RATIONAL HOMOTOPY
TYPE OF FUNCTION SPACES
Edgar H. Brown, Jr. and Robert H. Szczarba
Abstract. The main result of this paper is the construction of a minimal *
*model
for the function space F(X; Y ) of continuous functions from a finite typ*
*e, finite
dimensional space X to a finite type, nilpotent space Y in terms of minim*
*al models
for X and Y . For the component containing the constant map, ss*(F(X; Y )*
*) Q =
ss*(Y ) H-* (X; Q) in positive dimensions. When X is formal, there is a*
* simple
formula for the differential of the minimal model in terms of the differe*
*ntial of the
minimal model for Y and the coproduct of H*(X; Q). We also give a version*
* of the
main result for the space of cross sections of a fibration.
1. Introduction
In this paper we construct a minimal model in the sense of Sullivan [S] for
F(X; Y ), the space of continuous functions from a space X to a nilpotent space*
* Y ,
in terms of models for X and Y . We also generalize this to the space of sectio*
*ns of
a bundle. We first present some of the required background material and state t*
*he
main results of the paper.
Let Y be a connected, nilpotent CW complex (or simplicial set). A Q-localiza*
*tion
of Y consists of a space (or simplicial set) YQ and a mapping f : Y ! YQ such t*
*hat
ssj(YQ ) is a uniquely divisible group for all j > 0 and f* : Hj(Y ; Q) ! Hj(YQ*
* ; Q)
is an isomorphism for j > 0. (See [Hi].)
Let p : E ! X be a nilpotent fibration with connected fibre. Recall that a
Q-localization of p consists of a fibration p : E ! X and a fibre preserving m*
*ap
f : E ! E covering the identity map on X and defining a Q localization on the
fibres.
For any graded vector space V , we denote by Q[V ] the free commutative (in *
*the
graded sense) differential graded algebra generated by V . If v1; : :;:vn is a*
* basis
for V , we write Q[v1; : :;:vn] for Q[V ]. A commutative differential graded al*
*gebra
Typeset by AM S-T*
*EX
1
2 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
is said to be free, nilpotent, and of finite type (FNF) if as an algebra, A ' Q*
*[V ]
where V = {V q}, the dimension of V q is finite and V has a basis v1; v2; : :*
*s:uch
that dvn+1 2 Q[v1; : :;:vn] for all n. We say that A is minimal if A is FNF, dv*
* is
decomposable for all v 2 V , and V q= 0 for q 0.
All algebras considered in this paper will be commutative in the graded sens*
*e,
will have a unit 1, and all algebra mappings between algebras will preserve the
unit. We will consider Q as a subalgebra of any algebra A under the identificat*
*ion
r 7! r . 1.
In [S], Sullivan associates to each CW complex X a "rational deRham complex"
((X); d) whose homology is isomorphic to the singular cohomology of X with
coefficients in Q. He also associates to each nilpotent finite type CW complex*
* Y
a minimal algebra A and a homology isomorphism ff : A ! (Y ). The algebra A
determines the rational homotopy type of Y , is unique up to isomorphism, and
ssn (X) = Hom(ssn(X) Q; Q);
is isomorphic to the space of indecomposables in An. (For a nilpotent group G, *
*we
define G Q to be the direct sum (i=i+1) where {i} is lower central series
of G.) Thus, A = Q[V ] with V ' ss*(Y ). The minimal algebra A is called the
minimal model of X.
Before proceeding with the statements of our results, we describe an algebra*
*ic
construction which will be useful in what follows. Let A = Q[V ] be a FNF algeb*
*ra,
B a DG algebra such that Bq is finite dimensional for all q and Bq = 0 for q < *
*0.
Let B* the graded coalgebra with
Bq = Hom(B-q ; Q):
The differential on B* is the adjoint of the differential d on B and the coprod*
*uct
D : B* ! B* B* is the adjoint of multiplication. Let Q[A B*] be the free DG
algebra generated by the graded vector space A B* with differential d induced
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 3
by the tensor product differential on A B* and let I be the ideal in Q[A B*]
generated by 1 1 - 1 and by all elements of the form
0| 0 00
(1.1) a1a2 fi - (-1)|a2|.|fij(a1 fij)(a2 fij )
a1; a2 2 A, fi 2 B and
X
Dfi = fi0j fi00j
Then dI I (see Theorem 3.3) so that the differential d on Q[A B*] defines a
differential d on Q[A B*]=I.
Let Q[V B*] be the subalgebra of Q[A B*] defined by the inclusion V A =
Q[V ] and let ae be the composite
ae : Q[V B*] Q[A B*] ! Q[A B*]=I:
Theorem 1.2. The mapping ae : Q[V B*] ! Q[A B*]=I is an isomorphism of
graded vector spaces.
This is proved in Section 3 as part of Theorem 3.5.
Let d1 be the differential on Q[V B*] given by d1 = ae-1 dae. By definitio*
*n,
d1(v fi) can be computed by considering v fi as an element of Q[A B*],
computing d(v fi) in Q[A B*],
d(v fi) = (dv) fi + (-1)|v|v dfi;
then using the relations (1.1) to express (dv) fi as an element of Q[V B*]. F*
*or
example, if dv = v1v2; v1; v2 2 V , then
(dv) fi= (v1v2) fi
0| 0 00
= (-1)|v2|.|fij(v1 fij)(v2 fij )
P
where Dfi = fi0j fi00j. In particular, we see that the differential d1 depen*
*ds on
the differentials in A and B* and the coalgebra structure in B* (or equivalentl*
*y,
the algebra structure in B).
For any algebra E; (E) will denote the rational simplicial form of E. (See
Section 2 below or [S].)
4 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
Theorem 1.3. Let X and Y be CW complexes with Y nilpotent and finite type.
Let A and B be DG algebras as above with ff : A ! (Y ) a minimum model
for Y; A = Q[V ], and fi : B ! (X) a homology isomorphism. Then the space
|(Q[A B*]=I; d)| and hence |(Q[V B*]; d1)| is homotopy equivalent to the
function space F(X; YQ ).
The proof of the result is given in Sections 2 and 3. A version of Theorem 1*
*.3,
with a more complex description of d1, is proved by Haefliger in [H].
The following is a stronger version of Theorem 1.3 which is proved in Sectio*
*n 5.
Theorem 1.4. Let X; Y; A, and B be as in Theorem 1.3. Then there is a differ-
ential d2 on Q[V H*(B*)] such that
|(Q[V H*(B*)]; d2)|
is homotopy equivalent to F(X; YQ ).
We note that the Q-localization of a component of F(X; Y ) occurs as a compo-
nent of F(X; YQ ). ([Hi], Theorem 3.11).
We next state an analogue of Theorem 1.4 for the components of F(X; YQ ).
Theorem 1.5. Let Y be a nilpotent space of finite type and let X be a space of
finite type with Hq(X; Q) = 0 for q > N, some N. For f : X ! Y , let F(X; Y; f)
be the component of the function space F(X; Y ) containing f. Then F(X; Y; f) is
nilpotent, finite type, and has a minimal model (Q[W ]; d) where
!
X
W q= ssn (Y ) Hn-q (X; Q) =Kq
n
for subspaces Kq; q > 0. If f is the constant map, then Kq = 0.
Corollary 1.6. The homotopy group ssq(F(X; Y; f)) Q; q > 0, is isomorphic to
!
X
(ssn (Y ) Q) Hn-q (X; Q) =Kq
n
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 5
If f is the constant map, Kq = 0.
In Section 8 we give an example to show that the homotopy groups ssq(F(X; Y;*
* f))
Q can vary with f. Another example was given by G.W. Whitehead in the appendix
of [W].
The results stated above have analogues for the space of sections in a nilpo*
*tent
fibre space. For example, we have the following.
Theorem 1.7. Let p : E ! X be a nilpotent finite type fibration with fibre Y *
*and
Hq(X; Q) = 0 for q > N some N. Let f : X ! E be a section and denote by
(p; f) the component of the space of sections (p) containing f. Then (p; f) has
a minimal model A ' Q[W ] where W q; q > 0, is isomorphic to a quotient of
X
ssn (Y ) Hn-q (X; Q):
n
Corollary 1.8. The homotopy group ssq((p; f)) Q is isomorphic to a quotient
of
X
(ssn(Y ) Q) Hn-q (X; Q):
n
The proofs of the results for the space of sections are straightforward exte*
*nsions
of those for function spaces. An outline is given in Section 4. In particular, *
*a model
for E ! X takes the form B[V ] = B Q[V ] which replaces A in the above and the
ideal I is modified by adding the relations b fi - (-1)ff(|b|)fi(b); b 2 B; fi*
* 2 B*.
The computation of the differential in Q[V H*(B*)] is in general quite comp*
*li-
cated. Examples are given in Section 7 to illustrate this. However, if X is for*
*mal
[S], (that is, when the minimum model for (H*(X; Q); d = 0) is a minimal model
for X), the computation is much simpler as we describe below..
Suppose that X and Y are as in Theorem 1.5 and that X is formal with
Hq(X; Q) = 0 for q > N, some N. Let B = H*(X; Q) be the rational coho-
mology ring of X as a DG algebra with zero differential. Thus B* = H*(X; Q) is
the rational homology coalgebra of X. Let D* be the coproduct in H*(X; Q),
D* : H*(X; Q) ! H*(X; Q) H*(X; Q):
6 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
Applying the algebraic construction above, we obtain a DG algebra
(Q[V H*(X; Q)]; d1) where V ' ss*(Y ) and d1 = ae-1 dae with ae the mapping of
Theorem 1.2. At the end of Section 5 we prove
Theorem 1.9. Let X and Y be CW complexes with Y nilpotent and finite type,
X formal, and Hq(X; Q) = 0 for q > N, some N. Then the space
|(Q[ss*(Y ) H*(X; Q)]; d1)|
is homotopy equivalent to F(X; YQ ). Furthermore, given f : X ! Y , there is
an ideal K (see Section 6) depending on f, closed under d1 such that Q[V
H*(X; Q)]=K is FNF on positive dimensional generators and F(X; Y; f)Q is ho-
motopy equivalent to |(Q[V H*(X; Q)]=K; d2)|.
For example, if c 2 H*(X; Q) and v; v1; v2 2 V with dv = v1v2, then
d1(v c) = (-1)|v2|.|cj|(v1 c0j)(v1 c00j)
where D*c = c0j c00j. Note that, in this case, d1 depends on the differential i*
*n A
and the coproduct in H*(X; Q).
The proof of Theorem 1.9 is given at the end of Section 5. The special case *
*of
Theorem 1.9 when X = S1 was proved by Sullivan in [SV].
Remark 1.10. All of the above can be reformulated in terms of simplicial sets
instead of CW complexes. Then all the theorems remain true with only minor
modifications. In fact, step one in our proofs is to convert to simplicial sets.
Remark 1.11. In [BZ], we showed how to formulate real homotopy theory for
simplicial spaces in a way which is analogous to rational homotopy theory for s*
*im-
plicial sets. All of the above can be converted to real homotopy theory and the
theorems remain true if, for example, one starts with X = (B) and Y = (A)
where A and B are minimal algebras over the reals instead of the rationals.
Throughout this paper, the set of morphisms between objects S and T in a
specified category will be denoted by (S; T ), possibly with subscripts. For ex*
*ample,
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 7
if S and T are objects in the category of differential graded algebras, then (S*
*; T )
will denote the set of DG algebra mappings and (S; T )DG the set of mappings
preserving the grading and commuting with the differentials but not necessarily
preserving multiplication.
The paper is organized as follows. The proofs of the main results for funct*
*ion
spaces are given in Sections 2, 3, and 5. In Section 4, we outline the changes
necessary to extend our results on function spaces to the space of sections in a
nilpotent fibration. Section 6 deals with components and Section 7 contains two
examples. The first example shows that not all components of a function space
have the same Q-homotopy type. The second example shows that the formality
condition in Theorem 1.9 is necessary.
2. An Algebraic Model for F(X; Y )Q
For topological spaces U and W , let F(U; W ) be the space of all continuous
functions from U to W in the compact open topology. The goal of this section is*
* to
reduce the problem of determining the rational homotopy type of F(U; V ) to the
determination of the homotopy type of the rational simplicial form (Q[V B*]) of
the differential graded algebra Q[V B*] described in Section 1. Before proceedi*
*ng,
we recall some of the notions from rational homotopy theory that will be needed*
* in
what follows.
Let S denote the category of simplicial sets. (See [M] for a detailed accoun*
*t of
homotopy theory in S.) Recall that, for X; Y 2 S, the function space F(X; Y ) 2
S is defined by
F(X; Y )q = (X x [q]; Y )
where [q] is the simplicial set analogue of q with
[q]p = {(i0; i1; : :;:ip) : 0 i0 i1 : : :ip q}
and (X x [q]; Y ) is the set of simplicial mappings from X x [q] to Y . The face
and degeneracy operations in F(X; Y ) are defined in terms of the face inclusio*
*ns
8 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
[q - 1] ! [q] and degeneracy projections [q + 1] ! [q]. (See [M], p. 16,
for details.) For any X 2 S and y 2 X0 let Xy denote the simpicial subset of
X consisting of all simplicies whose vertices are all at y. When X is Kan, Xy *
*is
homotopy equivalent to the component of X containing y. For u 2 F(X; Y )0 =
(X; Y ), let F(X; Y; u) = F(X; Y )u.
Let A be the category of differential graded commutative, algebras and let p*
*q=
p(q) be the vector space of rational differential p-forms w on q:
w = ai1:::ipdti1: :d:tip
where the ai1:::ipare polynomials in t0; : :;:tq with rational coefficients. Th*
*en p =
{pq; q 0} is a simplicial set, q = {pq; p 0} is a DG algebra, and =
{pq; p; q 0} is a simplicial differential graded algebra. Define functors
: A ! S ; : S ! A
by (A) = (A; ) and (X) = (X; ). Then (A) is the rational simplicial form
of A and (X) the algebra of rational differential forms on X [BG].
For A; B 2 A, let F(A; B) 2 S be the function space defined by
F(A; B)q = (A; q B);
the space of all DG algebra mappings from A to qB. The simplicial structure on
F(A; B) is defined using the simplicial structure of . For w 2 F(A; B)0 = (A; B*
*),
let F(A; B; w) = F(A; B)(w).
We note that, in what follows, the function spaces F(X; Y ) and F(A; B) will*
* be
Kan [BZ] so that F(X; Y; u) and F(A; B; w) are actually homotopy equivalent to
the corresponding path components.
Our first result relates the function spaces for topological spaces and simp*
*licial
sets. Let (U) denote the singular complex of the space U and let |X| denote the
geometric realization of the simplicial set X.
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 9
Theorem 2.1. For X; Y 2 S, the simplicial set F(X; Y ) has the same homotopy
type as F(|X|; |Y |).
Proof. For f 2 F(X; Y )q; f : X x [q] ! Y , define mappings
F(X; Y ) -ff!F(|X|; |Y |) -fi!F(X; |Y |) -fl!F(|X|; |(|Y |)|)
in S as follows:
ff(f) = |f| : |X x [q]| = |X| x q ! |Y |
where |[q]| is identified with q.
For g 2 F(|X|; |Y |)q; g : |X| x q ! |Y | , fi(g) is the composite
fi(g) : X x [q] -oe!|X x [q]| = (|X| x q) -g-!|Y |
where oe : X x [q] ! |X x [q]| is the canonical inclusion.
For h 2 F(X; |Y |)q; h : X x [q] ! |Y |,
fl(h) = |h| : |X| x q ! |(|Y |)|:
It is now easily checked that the composite fiff is induced by the canonical
homotopy equivalence Y ! |Y | and that the composite flfi is induced by the
canonical homotopy equivalence |Y | ! |(|Y |)|. It follows that
ff* : ssq(F(X; Y )) ! ssq(F(|X|; |Y |)
is injective for all q. To see that ff* is surjective, consider y 2 ssq(F(|X|; *
*|Y |)) and
let x = (fiff)-1*fi*(y). Then
(flfi)*(ff*(x))= (fl*fi*)ff*(fiff)-1*fi*(y)
= fl*fi*(y)
so that ff*(x) = y since (flfi)* is an isomorphism. It follows that ff is a hom*
*otopy
equivalence.
10 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
We note that if X = U; Y = W for topological spaces U; W , then
F (U; W )' F (|U|; |W |)
' |F (|U|; |W |)| -|ff|'|F (U; W )|
As a consequence of Theorem 2.1, we can and will work in the category of
simplicial sets for the remainder of the paper.
Let X; Y 2 S, Y connected, Kan, nilpotent of finite type and Hq(X; Q) finite*
*ly
generated and zero for q > N for some N. Suppose u 2 F(X; Y )0 and h : Y ! YQ
is a Q localization.
Theorem 2.2. The mapping h* : F(X; Y; u) ! F(X; YQ ; hu) is a Q localization.
Proof. See [Hi], Theorem 3.11.
Let A; B 2 A; ff : A ! Y a minimal model for Y and fi : B ! X a homology
isomorphism. Assume further that Bq is finite dimensional for all q. Then
h : Y -j!Y -ff-!A
is a Q localization where j : Y ! Y is the canonical mapping and ff : Y !
A is induced by ff. [BZ] Thus, h* : F(X; Y; u) ! F(X; A; hu) is a Q localizatio*
*n.
Now, according to Theorem 2.20 of [BZ], there is a homotopy equivalence
fl : F(X; A) ! F(A; X):
Furthermore, according to this same theorem, the mapping
fi* : F(A; X) ! F(A; B)
is also a homotopy equivalence since fi is a homology equivalence. Thus we have
Theorem 2.3. Let "u= fi*fl(hu). Then F(X; Y; u) ! F(A; B; "u), is a Q localiza-
tion.
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 11
3. An Algebraic Reduction
We now proceed to analyze F(A; B) as described in Section 2. Let B* be the
differential, graded coalgebra with coproduct D and grading
Bq = Hom (B-q ; Q);
q 0, with differential the dual of the differential d on B. (We will also use *
*d to
denote the differential on A and on B* since no confusion seems likely.) Let {*
*bi}
be a basis for B and {fii} the dual basis for B*. For b 2 B, let |b| denote the*
* degree
of b and for any integer n, let ff(n) = [(n + 1)=2], the greatest integer in (n*
* + 1)=2.
Define
: (A B*; )DG ! (A; B)DG
in S by
X
(!)(a) = (-1)ff(|bi|)!(a fii) bi:
i
Note that the sum is finite since !(a fii) = 0 whenever degree (a fii) < 0 si*
*nce
is non negitatively graded and Bq is finite dimensional.
Theorem 3.1. The mapping is a simplicial isomorphism.
Proof. Before proving this result, we define two useful mappings. Let
" : Q ! B* B ; j : B B* ! Q
be the mappings defined by
X
"(1) = (-1)ff(|bi|)fii bi
i
j(b fi) = (-1)ff(|b|)fi(b)
Note that strictly speaking, "(1) is not an element of B B* since the sum may
be infinite. However, when " occurs in what follows, only a finite number of te*
*rms
will be involved.
12 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
Lemma 3.2. The mappings " and j above preserve grading and commute with
differentials.
Here we consider Q as the differential graded algebra with Q in degree zero *
*and
0 in degrees different from zero.
Proof. The first assertion is obvious. The second will be proved if we can show*
* that
d O " = j O d = 0. We begin by showing d O " = 0.
Let aji2 Q be defined by
X j
dbi = aibj
j
so that
X
dfii= aijfij:
j
Then
X i
d"(1) = (-1)ff(|bi|)(d*fii bi+ (-1)|b |fii dbi)
Xi X
i|j
= (-1)ff(|bi|)aijfij bi+ (-1)ff(|bi|)+|baifii bj:
i;j i;j
Interchanging the roles of i and j in the first sum and using the fact that aij*
*= 0
unless |bj| = |bi| + 1, we have
X i j j
d"(1) = (-1)ff(|bi|+1)+ (-1)ff(|bi|)+|bi|aifii bj:
i;j
The fact that this expression vanishes is a consequence of the easily verified *
*identity
[(n + 2)=2] = 1 + [(n + 1)=2] + n mod 2:
The equation j(d(b fi)) = 0 is proved in the same way. We leave the details to
the reader.
For u 2 (A; B)DG , let (u) be the composite
A B* -uid--! B B* -idj--! Q ' :
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 13
Note that (!) is the composite
A -! A Q -id"--!A B* B -!id--! B:
and that can be written more explicitly as
(u)(a fii) = (-1)ff(|bi|)!i
P
where u(a) = !j bj
j
Now, if u and ! preserve grading and differentials, then each of the mapping*
*s in
the definition of (u) and (!) also preserve grading and differentials. Thus we
have simplicial mappings
: (A; B)GD ! (A B*; )GD
: (A B*; )GD ! (A; B)GD
which satisfy
X
((u))(a) = (-1)ff(|bi|)(u)(a fii) bi
Xi
= !i bi = u(a)
i
and
((!))(a fii) = !(a fii)
P P
since (!)(a) = i!i bi = i(-1)ff(|bi|)!(a fii) bi. Thus each of the com-
posites O and O is the identity and Theorem 3.1 is proved.
Consider now Q[A B*], the free commutative DG algebra on the differential
graded vector space A B*. Each ! 2 (A B*; )DG extends to a unique DG
algebra mapping ! : Q[A B*] ! . Thus we can identify (A B*; )DG with
(Q[A B*]; ), the DG algebra mappings, and consider as a simplicial isomor-
phism
: (Q[A B*]; ) ! (A; B)DG :
Let oe : A A B* ! Q[A B*] be given by
X 0
oe(a1 a2 fi) = a1a2 fi - (-1)|a2|.|fij|(a1 fi0j)(a2 fi00j):
P j
where Dfi = fi00j fi000j.
14 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
Theorem 3.3. For ! 2 (Q[A B*]; ), the mapping (!) 2 (A; B)DG is a
DG algebra mapping if and only if !(1 1*) = 1 and !(oe(a1 a2 fii)) = 0 for
all a1; a2 2 A and all fi 2 B*. Furthermore, if I Q[A B*] is the ideal genera*
*ted
by 1 1* - 1 and oe(a1 a2 b), a1; a2 2 A; b 2 B*, then dI I.
Corollary 3.4. The mapping defines a simiplicial isomorphism
: (Q[A B*]=I; ) ! (A; B):
Proof. We first note that an element z 2 B is zero if and only if
(id j)(z fik) = 0 for all k.
Using the fact that
ff(n + m) = ff(n) + ff(m) + nm mod 2
for all non-negative integers n and m, a straightforward computation shows that
(id j)(( (!)(a1a2) - ( (!)(a1))( (!)(a2)) fik) = !(oe(a1 a2 fik))
which proves the first part of Theorem 3.3. To prove that dI I, consider the
mappings
: A A ! A
D : B* ! B* B*
: Q[A B*] Q[A B*] ! Q[A B*]
o : A A B* B* ! A B* A B*
where is the multiplication in A, D the comultiplication in B*, is the multi*
*pli-
cation in Q[A B*], and
0|.|fi| 0 00
o (a a0 fi fi00) = (-1)|a a fi a fi :
Then oe = idB* - o (idAA D) and, since each of the mappings on the right
commute with the corresponding differential, so does oe and the theorem is prov*
*ed.
Let A = Q[V ] where V qis finite dimensional for all q and let ae be the com*
*posite
ae : Q[V B*] Q[A B*] ! Q[A B*]=I:
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 15
Theorem 3.5. The mapping ae : Q[V B*] ! Q[A B*]=I is an isomorphism of
graded algebras.
Let d1 be the differential on Q[V B*] defined by d1 = ae-1 dae where d is t*
*he
differential on Q[A B*]=I induced by the tensor product differential on A B*.
We then have
Corollary 3.6. Let A = Q[V ]; B 2 A; ff : A ! Y a minimum model for Y , and
fi : B ! X a homology isomorphism. Assume further that Bq is finite dimen-
sional for all q. Then the simplicial set (Q[V B*]; d1) is homotopy equivalent
to F(X; YQ ).
The proof of Theorem 3.5 proceeds by constructing an inverse for ae. We begin
with some preliminaries.
Given v1; : :;:vk 2 V and fi` 2 B*, define T (v1; : :;:vk; fi`) in Q[V B*]*
* induc-
tively by
T (v1; : :;:vk; fi`)= v1 fi` ifk = 1;
X
= "kij`ijT (v1; : :;:vk-1 ; fii)(vk fij) ifk > 1
i;j
where
X
D(fi`) = `ijfii fij
"kij= (-1)|bi|(|bj|+|vk|):
Lemma 3.7. For any v1; : :;:vk 2 V; fi` 2 B*; 1 < m k, we have
X
T (v1; : :;:vk; fi`) = "mkij`ijT (v1; : :;:vm-1 ; fii)T (vm ; : :;:*
*vk; fij)
i;j
where
"mkij= (-1)|bi|(|bj|+|vm |+...+|vk|):
Proof. We proceed by induction on k. The case k = 2 follows immediately from the
definition of T . Suppose the result is true for k - 1 and let m satisfy 1 < m *
* k. If
16 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
m = k, the result again follows from the definition of T so we can assume 1 < m*
* < k.
Then, by definition, T (v1; : :;:vk; fi`) is given by
X
= "kij`ijT (v1; : :;:vk-1 ; fii)(vk fij)
i;j
X X
= "kij`ij "mk-1rsirsT (v1; : :;:vm-1 ; br)T (vm ; : :v:k-1; bs)(vk *
*fij)
i;j rs
by induction. Now `ijirs= isj`risince multiplication in B is associative. Making
this substitution and rearranging terms we can express T (v1; : :;:vm ; fi`) as
X X
"kij"mk-1rs`riT (v1; : :;:vm-1 ; fik) isjT (vm ; : :;:vk-1 ; bs)(v*
*k fij):
r;j s;j
Except for a factor "ksj, the second sum on the right is T (vm ; : :;:vk; fii) *
*by definition.
Inserting this factor twice, we have
X
T (v1; : :;:vk; fi`) = "kij"mk-1rs"ksj`riT (v1; : :;:vm-1 ; fik)T (vm ; *
*: :;:vk; fii):
r;i
Using the fact that isj= 0 unless |bs|+|bj| = |bi| and `ri= 0 unless |br|+|bi| *
*= |fi`|,
we see that |bj| = |br| mod 2 and |bs| = |br| + |bi| mod 2. An easy computa*
*tion
now shows that
"kij"mk-1rs"ksj= "kri
which gives the required result
Lemma 3.8. For any v1; : :;:vk 2 V; fi` 2 B*; 1 m < k, we have
T (v1; : :;:vk; fi`) = (-1)|vm |.|vm+1T|(v1; : :;:vm-1 ; vm+1 ; vm ; : :;*
*:vk; fi`):
Proof. For k = 2 we have
X
T (v1; v2; fi`)= "2ij`ij(v1 fii)(v2 fij)
i;j
X i j
= "2ij(-1)(|v1|+|b |)(|v2|+|fij|)+|fii|.|b`|ji(v2 fij)(v1*
* fii)
i;j
X
= "1ji(-1)|v1||v2|`ji(v2 fij)(v1 fii)
= (-1)|v1||v2|T (v2; v1; fi`):
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 17
If k > 2 and m < k - 1, the result follows by induction. If k > 2 and m = k - 1,
one can use Lemma 3.7 to reduce the result to the case k = 2. We leave the deta*
*ils
to the reader.
Now to the proof of Theorem 3.5. To begin with, a straightforward induction
shows that ae is surjective. Indeed, any element v1 : :v:k fi` can be expressed
mod I as a sum of products of terms of the form v1 : :v:m fii; m < k. To pro*
*ve
ae injective, we define
oe : Q[A B*] ! Q[V B*]
by oe(v1 : :v:k fi`) = (-1)ff(|b`|)T (v1; : :;:vk; fi`). Then oe is well define*
*d by Lemma
3.8, oe(I) = 0 by Lemma 3.7, and oeae = identity by inspection. Thus ae is bije*
*ctive
and the theorem is proved.
4. An Algebraic Model for (p)
In this section, we outline the modifications required to prove the analogues
of the results of the previous two sections for the space of sections of a nilp*
*otent
fibration. We begin with an analogue of Theorem 1.2.
Suppose p : E ! X is a fibration in S and let (p) be the simplicial form of
the space of sections of p, that is,
(p)q = {f : [q] x X ! E : pf = p2}
where p2 is the projection of [q] x X onto X. We say that p : E ! X in S is
nilpotent and of finite type if there is a sequence En ! En-1 of principal fibr*
*ation
with E1 = X and with group and fibre K(Gn; mn) satisfying
(i) Gn is a finitely generated abelian group, mn mn+1 , and mn ! 1 as
n ! 1.
(ii)There is an isomorphism E ! limEn such that the diagram
E ----! lim-En
?? ?
y p ?y
X ---id-! X = E1
18 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
is commutative.
Let B be a graded algebra and V a graded vector space. We denote by B[V ] the
graded algebra B Q[V ]. A straightforward generalization of Theorem 2.6 of [BZ]
yields the following
Theorem 4.2. Let p : E ! X be a nilpotent fibration as above and let B 2 A and
f : B ! (X) be a homology isomorphism. For each n = 1; 2; : :;:let V (n) be the
graded vector space with
V (n)mn = Hom(Gn; Q)
V (n)j= 0; j 6= mn:
P1 Pn
Set V = V (j) and V [n] = V (j). Then there is a differential d on A = B[*
*V ]
j=1 j=1
and a homology isomorphism h : A ! (E) such that d(V (n)) B[V [n - 1]], dv
is decomposable for v 2 V and the diagram
B ---f-! (X)
?? ?
y ?y
A ---h-! (E)
is commutative.
We next define an analogue of the space of sections in the category A. Let
A = B[V ] be as in Theorem 4.1 and i : B ! A the inclusion. Then (i) is the
simplicial set defined by
(i) = {u 2 F(A; B) : u(b) = 1 b forb 2 B}:
Recall that F(A; B) 2 S is the function space defined by
F(A; B)q = (A; q B)q:
The proof of Theorem 2.20 in [BZ] then carries over to produce
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 19
Theorem 4.3. If i : B ! A is as above, then the components of (i) include the
Q-localizations of the components of (p).
We now identify (i) with a simplicial subset of (Q(A B*)=I) F(A; B).
Recall that in Section 3 we defined an isomorphism : (A B*; q) ! (A; q B)
given by
X
(w)(a) = (-1)ff(|bi|)w(a fii) bi:
i
We need to determine which w satisfy (w)(b) = 1 b, that is
X
(w)(bj) = (-1)ff(|bj|)w(bj fii) bi = 1 bj
i
or, equivalently,
w(bj fii) - (-1)ff(|bj|)fii(bj) = 0
for all i and j. Let J Q[A B*] be the ideal generated by I and all elements of
the form b fi - (-1)ff(|b|)fi(b); b 2 B; fi 2 B*.
Theorem 4.4. The mapping induces an isomorphism
(Q[A B*]=J) (i):
We alter the proof of Theorem 3.5 slightly to prove
Theorem 4.5. The inclusion V B(V ) = A induces an isomorphism
Q[V B*] Q[A B*]=J:
Proof. Note that A B* is spanned by elements of the form bv1v2 . .v.` fi, b 2
B; v1; : :;:v` 2 V; fi 2 B*.
Define
T (b; v1; v2; : :;:vk; fi`) = fi`ijT (v1; v2; : :;:vk; fij)
where T (v1; v2; : :;:vk; fij) is as in the proof of Theorem 3.5 and
fi`ij= fii(b)`ij(-1)"
where " = ff(|bi|) + (|vt|)|bi|. The proof of Theorem 4.5 then proceeds exactly*
* as
in Theorem 3.5.
20 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
5. A Reduction of Q[V B*] to Q[V H*(B*)]
Suppose A = Q[V ] and B are as in Corolary 3.6. Let {vi} be a basis for V su*
*ch
that |vi| > 0; |vi| |vi+1| and dvi+1 2 Q[Vi] where Vi is the subspace spanned *
*by
v1; : :;:vi. Note that dvi is decomposable and |vi| ! 1 as i ! 1. Let {ai; bi; *
*ci}
be a basis for B* with d*ai = bi and d*ci = 0.
Lemma 5.1. The algebra Q[V B*] has a free set of generators wij; uij and vij
satisfying
(i) wij= vi cj + xij where xij2 Q[Vi-1 B*]
(ii)uij= vi aj and vij= duij
(iii)dwij2 Q[{wk` : k < i}]
(iv) If d(vi cj) is decomposable so is dwij.
Let W be the subspace of Q[V B*] spanned by {wij} and U the subspace
of Q[V B*] spanned by {uij; vij}. Thus W is isomorphic to V H*(B*) and
Q[V B*] is isomorphic to Q[W ] Q[U].
Lemma 5.2. The algebra Q[W ] is a deformation retract (in the category A) of
Q[V B*].
(See [BG], Section 4 for a discussion of homotopy theory in the category A.)
Before proving these lemmas, we state the following consequence.
Theorem 5.3. There is a differential on Q[V H*(B*)]( Q[wij]) making it a
FNF (as in section 1) algebra such that
Q[V B*] Q[V H*(B*)] Q[{uij; vij}]
as DGA algebras (where duij= vij). Furthermore Q[V H*(B*)], with this differ-
ential, is a deformation retract of Q[V B*] in the category A. Thus, (Q[V B*])
has the homotopy type of (Q[V H*(B*)]).
Proof of Lemma 5.1. We use induction on i. Let w1j = v1 cj; u1j = v1 aj and
v1j = du1j. Then dwij = 0 since dv1 = 0 and conditions (i),...,(iv) are satisf*
*ied.
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 21
Suppose wk;`have been defined for k < i and satisfy (i),...,(iv). Using induct*
*ion
and the fact that
vk`= d(vk a`)
= vk b` + (dvk) a`
= vk b` mod Q[Vk-1 B*]
we have wk` = vk c`; uk` = vk a` and vk;`= vk b`, modulo Q[Vk-1 B*].
Since vk c`; vk a` and vk b`; k < i freely generate Q[Vi-1 B*], it follows *
*that
wk;`; uk;`and vk`; k < i freely generate Q[Vi-1 B*]. Hence, we have
Q[Vi-1 B*] = Q[wk` : k < i] Q[uk`; vk` : k < i]
as DG algebras. Since
d(vi cj) = (dvi) cj 2 Q[Vi-1 B*]
is a cycle,
d(vi cj) = x + du
where x 2 Q[wjk : j < i] and u 2 Q[Vi-1 B*]. Let wij = vi cj - u. Then
dwij= x as specified by 5.1 (iii).
Suppose d(vi cj) is decomposable and dwij= x as above. Then
X
x = ffik`wk
k 1. Finally, if u = 0 and dw is decomposable
for all w 2 W , then E=Ku is minimal and W"1 = W 1.
Proof. Note first of all that if w 2 (E) vanishes on Ku, then, for w 2 W 0, we
have
!(w) = !(w - u(w) + u(w))
= u(w)
24 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
since u(w) 2 Q. Thus, u is the only 0-simplex in (E=Ku) so that (E=Ku)
(E)u.
To prove that dKu Ku, it is enough to prove that d! 2 Ku for ! in each of
the three sets of generators defining Ku. If w 2 E-1 , then
dw = dw - u(dw)
= w - u(w ) 2 Ku
since u(dw) = du(w) = 0. We next prove E=Ku = Q[W" ]. Let K = Ku and let
Ln A be the ideal generated by the set
{xn : m < n and |xn| < 0} [ {xn - u(xn) : m < n and |xn| = 0}:
Then, if |xn| = 0,
X
dxn = anm xm mod Ln;
|xmm|=1 1] Q[xm : |xm | = 1; xm 6= dxn ]
Furthermore, the differential of a generator in E=K only involoves previous gen*
*er-
ators so that E=K is FNF.
We next prove E ! E=K induces a homotopy equivalence of (E=K) into
(E)u. Let En = Q[Wn]. It is sufficient to prove that En ! En=K induces a
homotopy equivalence since (E) ! (En) is a fibration with fibre whose connec-
tivity increases with n. We proceed by induction on n. The case n = 0, is immed*
*i-
ate. Suppose it is true for n - 1. Let un = u|(En) and let Kn be the ideal K for
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 25
En. If |xn| < -1, then (E=Kn) = (En-1 =Kn-1 ) and (En)un = (En-1 )un-1
by definition. If |xn| = -1, then (En=Kn) = (En-1 =Kn-1 ) by definition. If
! 2 (En-1 )un-1 and |w| = q, then d!(dxn) = 0 and @q0w(dxn) = u(dxn) = 0 so
that w(dxn) = 0. We then have
(En-1 )un-1 = {! 2 (En-1 )un-1 : !(dxn) = 0}
= (En)un :
Suppose |xn| 0. We then have fibrations p1 and p2 in the following commuta-
tive diagram with Q[xn] the fibre of both.
(En-1 =Kn-1 [xn]) ----! {! 2 (En) : ! | En-1 2 (En-1 )un}
? ?
p1?y p2?y
(En-1 =Kn-1 ) ----! (En-1 )un-1
For |xn| > 0 this gives a homotopy equivalence (En=Kn) ! (En)un . If |xn| = 0,
then p1 and p2 are covering projections. Let v 2 ((En-1 =Kn-1 )[xn])0 be defined
by v(xn) = u(xn). Then v maps to un above and, passing to components, we have
a homotopy equivalence
(En-1 =Kn-1 [xn])v ! (En)un :
Thus it is sufficient to show that
(En=Kn) ! (En-1 =Kn-1 [xn])v
is a homotopy equivalence. If dxn 2 Kn-1 , the two sides are equal since d!(xn)*
* = 0
and @q0! = un implies !(xn) = u(xn). Suppose dxn =2Kn-1 . Then, as in the proof
of freeness above, we may assume dxn = xm ; m < n. Let x = xn and y = xm . If
j < i; dxj = ffj + yfij in En-1 =Kn-1 and ffj; fij 2 Ej-1. Let
xj = xj - xfij j 1; j 6= m:
One can solve for the xi's in terms of xj's and hence dxj 2 Q[xj : i < j]. Thus
En-1 =Kn-1 [xn] = Q[xj] Q[x; y];
26 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
as DG algebras. Hence Q[xj] is a deformation retract of En-1 =Kn-1 [xn] as in
Lemma 5.1. The image of xj under the map
Q[xj] ! En-1 =Kn-1 [xn] ! En=Kn
is xj - u(x0)fij. Since fij is a polynomial in xk; k < j, this map is an isomor*
*phism.
Hence
(En=Kn) ! (En-1 =Kn-1 [xn])v ! (Q[xj])
are homotopy equivalences.
Corollary 6.2. If E = (Q[W ]; d) is as in Theorem 6.1 and u 2 (E)0, there is a
W and d such that W is isomorphic to a quotient of W , E = (Q[W ]; d) is min*
*imal
and (E ) is homotopy equivalent to (E)u.
Proof. In [BG], Propositions 7.11 shows that such an E exists which is homotopy
equivalent to E"as in Lemma 5.2.
7. Two Examples
In this section we present two examples. In the first, we determine the set
of components and describe a minimal model for each component. It follows from
this description that not all components have the same homotopy type. The second
example shows that the formality condition is necessary in Theorem 1.9.
Let A = Q[V ] where V has a basis {x; y; v} with
|x| = 2 ; dx = 0;
|y| = 4 ; dy = 0;
|v| = 5 ; dv = xy:
Let B* = Q[ff; fi] be the Hopf algebra with ff; fi primitive and with |ff| = -2*
*; |fi| =
-3, and dff = dfi = 0. Let W = V B* and let Q[W ] be the DG algebra with
differential induced by the differential in Q[A B*] as in Corollary 3.6. Then *
*Q[W ]
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 27
in dimension q = -1; 0; 1; 2 has a basis as follows. (We write xff for x ff a*
*nd x
for x 1.)
degree - 1 : xfi; yfffi; vff3
degree 0 : xff; yff2; vfffi
degree 1 : yfi; vff2
degree 2 : x; yff; vfi
Ignoring terms involving negative dimensional generators, the non zero differen*
*tials
of the basis above are given by
d(vff3)= 3(xff)(yff2)
d(vfffi)= (xff)(yfi)
d(vff2)= 2(xff)(yff) + x(yff2)
d(vfi)= x(yfi)
For example,
d(vfffi)= d(v fffi)
= xy fffi
= (x 1)(y fffi) + (x fffi)(y 1)
+ (x ff)(y fi) + (x fi)(y ff)
using equation 1.1 since
D(fffi)= D(ff)D(fi)
= (1 ff + ff 1)(1 fi + fi 1):
However, the first, second, and fourth terms involve negative dimensional gener*
*ators
so that
d(vfffi)= (x ff)(y fi)
= (xff)(yfi):
28 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
The other cases are similar.
We now compute the set of components of (Q[W ]). Note first of all that the
set (Q[W ])0 of 0-simplices can be identified with the set of mappings u : W 0!*
* Q
satisfying u(dv) = 0 for any v 2 W -1. Thus, u is determined by a; b; c 2 Q whe*
*re
u(xff) = a; u(yff2) = b; u(vfffi) = c:
The condition u(dv) = 0 implies that ab = 0 so that
(Q[W ])0 {(a; b; c) 2 Q3 : ab = 0}:
Let ua;b;cdenote the element of (Q[W ])0 corresponding to (a; b; c).
In the same way, it can be shown that the set (Q[W ])1 of 1-simplices of
(Q[W ]) consists of all pairs (u; s) where
u : W 0! Q[t]
s : W 1! Q[t; dt]
satisfy u(dw) = 0 for w 2 W -1 and s(dz) = du(z) for z 2 W 0. Suppose that
u(xff) = a(t)
u(yff2)= b(t)
u(vfffi)= c(t)
s(yfi)= f(t)dt
s(vff2)= g(t)dt:
Then,
0 = u(d(xff))
= du(xff) = ff0(t)dt
so a(t) = a is a constant. Similarly, b(t) = b is a constant, ab = 0, and c0(t)*
* = af(t).
Thus (Q[W ])1 can be identified with the set of 5-tuples (a; b; c(t); f(t); g(t*
*)) where
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 29
a; b 2 Q; c(t); f(t); g(t) 2 Q[t]. Note that, with this identification, @0(u; s*
*) is the zero
simplex corresponding to (a; b; c(0)) and @1(u; s) is the zero simplex correspo*
*nding
to (a; b; c(1)).
Suppose that (u; s) is as above and a = 0. Then c0(t) = 0 and c(t) = c is
constant. It follows that @0(u; s) = @1(u; s) so that the component of (Q[W ])
containing the zero simplex u0;b;ccontains no other 0-simplices. Thus, each u0;*
*b;c
determines a distinct component.
Next, suppose that a 6= 0 and b = 0 and that ua;0;0; ua;0;c2 (Q[W ])0. Let
(u; s) 2 (Q[W ])1 correspond to (a; 0; c(t); c0(t)=a; 0) where c(0) = 0 and c(1*
*) = c.
Then @0(u; s) = wa;0;0; @1(u; s) = wa;0;cso that each 0-simplex wa;0;clies in t*
*he
same component as wa;0;0.
It follows from the discussion above that ss0((Q[W ])), the set of components
of (Q[W ]), is given by
ss0((Q[W ])) Q2 [ {Q - (0)}:
Let u = u0;0;0; w = u1;0;02 (Q[W ])0. A straightforward computation shows
that the ideal Ku (defined in Section 6) is generated by all z 2 Q[W ] with |z|*
* < 0
together with the elements xff; yff2; vfffi. Similarly, the ideal Kw is gener*
*ated by
all z 2 Q[W ] with |z| < 0 together with xff - 1; yff2; vfffi; yfi. Set Au = Q[*
*W ]=Ku
and Aw = Q[W ]=Kw . According to Theorem 6.1, the component of (Q[W ])
containing u is homotopy equivalent to (Au) and the component of (Q[W ])
containing w is homotopy equivalent to (Aw ). Now, A1uhas basis yfi; vff2 with
d(yfi) = d(vff2) = 0 and A1whas basis vff2 with d(vff2) = 2yff. It follows that
H1((Q[W ])u) ' Q2
H1((Q[W ])w ) ' 0
so that the components (Q[W ])u and (Q[W ])w have different homotopy types.
We note that Au is a minimum model for (Q[W ])u. Applying Theorem 6.1,
30 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
we see that the subalgebra of Aw generated by x; vfi; v - x(vff) + (x)2(vff2) *
*is a
minimum model for (Q[W ])w .
We next calculate differentials in a nonformal example. This yields a result
different from what is obtained using the coproduct in H*(B*) as in the formal
case.
Let A = Q[V ] where V has a basis {v1; v2; v3; v4} with
|v1|= 3 ; dv1 = 0
|v2|= 2 ; dv2 = 0;
|v3|= 4 ; dv3 = v1v2;
|v4|= 6 ; dv4 = v1v3:
Let B = Q[x; y; z; u; w] where U has a basis {x; y; z; u; w} with
|x| = |y| = |z| = z; dx = dy = dz = 0;
|u| = |w| = 3; du = xy; dv = yz;
and consider the sets
fl= {x; y; z; x2; y2; z2; xz; oe = uz - xw};
ff= {u; v; xu; yu; xw; yw; zw};
fi= {xy; yz}:
Then ff[fi [fl is a basis for B1 . .B.5and we let ff*[fi* [fl* be the dual basi*
*s for
(B1 . . .B5)*. The basis ff* [ fi* [ fl* is of the kind required in Section 5.*
* More
explicitly, the elements of fl* correspond to the ci, the elements of ff* corre*
*spond
to the ai, and the elements of fi* correspond to the bi.
We now construct the elements wij2 Q[V B*] satisfying (i) and (iii) of Theor*
*em
5.1. Let wij= vi cj, i; j = 1; : :;:4; cj 2 fl* except for cj = oe*; i = 3; 4.*
* Then (i)
and (iii) of Theorem 5.1 are satisfied for the wij and we need only define
w1 = v3 oe* mod Q[V2 B*];
w2 = v4 oe* mod Q[V3 B*];
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 31
with dw1 and dw2 satisfying (iii). A straightforward computation shows that
Doe* = 1 oe* + oe* 1 + u* z* + z* u*
so that
d(v3 oe*) = v1v2 oe*
= (ff1 oe*)(v2 1*) + (v1 1*)(v2 oe*)
+ (v1 z*)(v2 u*) + (v1 u*)(v2 z*):
The first two terms in this expression are of the required form but the last tw*
*o are
not. To eliminate the last two terms, we note that
d((v2 z*)(v2 (xy)*) + (v1 (xy)*)(v2 z*))
= -(v1 z*)(v2 u*) - (v1 u*)(v2 z*):
Defining w1 by
w1 = v3 oe* + (v1 z*)(v2 xy*) + (v1 xy*)(v2 z*)
we see that dw1 has the required form.
In the same way, we define w2 by
w2 = v4 oe* + (v1 z*)(v3 xy*)
+ (v1 xy*)(v3 z*):
Then
dw2 = (v1 oe*)(v3 1*) + (v1 1*)(w1)
- (v1 z*)(v1 x*)(v2 y*)
- (v1 z*)(v1 y*)(v2 x*)
which has the required form.
32 EDGAR H. BROWN, JR. AND ROBERT H. SZCZARBA
On the other hand, if "oeis the element of H*(B*) determined by oe*, then
"D"oe= "oe 1 + 1 "oe
where D" is the coproduct in H*(B*). We then have
d(v4 "oe)= v1v3 "oe
= (v1 "oe)(v3 1) + (v1 1)(v3 "oe):
ON THE RATIONAL HOMOTOPY TYPE OF FUNCTION SPACES 33
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*al homo-
topy type, Mem. Amer. Math. Soc. No. 179 (1976).
[BZ] E.H. Brown and R.H. Szczarba, Continuous cohomology and real homotopy t*
*ype, Trans.
AMS 311 (1989), 57-106.
[H] A. Haefliger, Rational homotopy of the space of sections of a nilpotent*
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