RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS OF
COMPLEXES WITH FINITELY MANY HOMOTOPY GROUPS
SAMUEL B. SMITH
Abstract. For simply connected CW complexes X with finitely many, finitely
generated homotopy groups1, the path components of the function space M (X, X)
of free self-maps of X are all of the same rational homotopy type if and only if
all the k-invariants of X are of finite order. In case X is rationally a two-stage
Postnikov system the space M0 (X, X) of inessential self-maps of X has the
structure of rational H-space if and only if the k-invariants of X are of finite
order.
1. Introduction. Given spaces X and Y let M (X, Y ) denote the space of
free, continuous maps from X to Y with the compact-open topology. Given a
map f : X → Y, we let Mf (X, Y ) denote the component of f in M (X, Y ). In
particular, we denote by M0 (X, Y ) the space of inessential maps from X to Y and
by M1 (X, X) the space of self-maps of X which are homotopic to the identity. If X
and Y are spaces with base points then we have the subspace M (X, Y )∗ of M (X, Y )
consisting of all based maps between X and Y. Given a based map f : X → Y, we
write Mf (X, Y )∗ for the component of f in M (X, Y )∗ .
In his 1956 paper [20], R. Thom used a Postnikov decomposition of Y to show
that the homotopy groups of M (X, Y ) are determined up to a series of group extensions involving the cohomology of X with coefficients in the homotopy of Y .
To illustrate his technique, Thom computed the rational homotopy groups of certain function spaces. He also showed how by analyzing the evaluation map it
1991 Mathematics Subject Classification. Primary 55P62, 55P15, 58D99.
Key words and phrases. Function Spaces, Minimal Model, Postnikov Tower, Rational Homotopy Equivalence, Hirsch Lemma.
1 Here and throughout, this hypothesis may be replaced with the assumption that X is a
simply connected finite CW complex with finite-dimensional rational homotopy. See the remark
after Proposition 2.1 below.
1
2
SAMUEL SMITH
is possible, in certain cases, to determine the rational homotopy type of components of M (X, Y ). While predating by many years Sullivan’s development of the
theory of minimal models, Thom’s work established the techniques for studying
function spaces within the framework of rational homotopy theory. In [7], Haefliger
completed a program initiated by Sullivan [18] with the construction of a rational
homotopy-theoretic model for the space of sections of a nilpotent bundle. The construction embodied the two aspects of Thom’s approach to the study of function
spaces – Haefliger and Sullivan’s algebraic model mirrored Thom’s geometric use
of Postnikov decompositions and, as Thom’s work suggested, a key element of the
construction was the algebraic characterization of the relevant evaluation maps.
In this paper we apply Thom’s methods within the context of Sullivan’s minimal
models to study the rational homotopy types represented by the components of
function spaces.
A basic problem concerning the function space M (X, Y ) is to classify its components up to homotopy type. In case X is an n-manifold and Y an n-sphere, Hansen
[8] classifies the components in terms of the degree of the map from X to Y . Møller
obtains an analogous classification theorem in [14] when X and Y are taken to be
complex projective spaces. If Y is an H-space with homotopy inverses, it is easy
to show that the components of M (X, Y ) represent one homotopy type. Our first
main result states that when X = Y the assumption that X is a rational H-space
is necessary to ensure the components of M (X, X) are of one rational homotopy
type. We prove
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
3
Theorem 1. If X is a simply connected complex with finitely many, finitely generated homotopy groups, the path components of M (X, X) are all of the same rational
homotopy type if and only if the k-invariants of X are all of finite order.
The proof of Theorem 1 is computational; we use rational Postnikov towers together with Thom’s method for calculating homotopy groups of function spaces,
which we recall in §3. In fact, the key lemma (Lemma 3.2) is the calculation of the
rational homotopy groups of the space M0 (X, Y ) for X and Y simply connected
spaces of finite homotopy type. Our computational method also permits the determination of the rational homotopy type of the space M1 (X, X) for certain spaces
X, which has implications for the theory of classifying fibrations (see §5). As an example we determine the rational homotopy type of M1 (X, X) when X is a product
of spheres ( Theorem 2).
If X is an H-space then multiplication of maps gives the spaces M0(X, X)∗ and
M0 (X, X) the structure of H-space also. In Lemma 6.1, we prove that if X is a twostage Postnikov system, then the space of based inessential maps M0 (X, X)∗ has
the structure of H-space regardless of whether or not X does. This fact contrasts
sharply with the situation for the space of free inessential maps. In fact, our second
main result implies that if X is rationally a two-stage Postnikov system, then the
assumption that X is a rational H-space is necessary to ensure that M0(X, X) has
the structure of rational H-space. We prove
Theorem 3. Let X be a simply connected complex with finitely many, finitely
generated homotopy groups whose rationalization is a two-stage Postnikov system.
Then there is a rational equivalence
M0 (X, X) 'Q X × M0 (X, X)∗ .
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SAMUEL SMITH
The proof of Theorem 3 consists in constructing the minimal model for M0 (X, X).
To accomplish this we use the Hirsch Lemma [4, Lemma 3.1] and the algebraic characterization of certain evaluation maps given by Haefliger [7], which we recall in §7.
Our proof follows the approach taken by Møller and Raussen in [15]. We remark
that since M1 (X, X) always has the structure of H-space, with multiplication given
by composition of maps, Theorem 3 can be viewed as a refinement of Theorem 1
for two-stage Postnikov systems.
This paper represents a portion of the author’s Ph.D. thesis. The author would
like to thank his thesis advisor, Professor Donald Kahn, for his assistance and
encouragement during the preparation of this work.
2. Preliminaries. In this section we recall results and establish notation for use
in the sequel. We begin with a result on
Function Spaces and Rationalization. The space M (X, Y ) does not, in general,
have the homotopy type of a CW complex. (For an example of this failure, see
[13, page 273].) Since we wish to consider the rationalization of components of
M (X, Y ), we must restrict the spaces X and Y under consideration so that the
components of M (X, Y ) are nilpotent CW complexes. The following proposition
provides us with a suitable class of spaces with which to work:
Proposition 2.1. Let X and Y be simply connected countable CW complexes with
either
a) X a finite complex or
b) X having finitely generated homotopy groups and Y having finitely many,
finitely generated homotopy groups.
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
5
Then each component Mf (X, Y ) of M (X, Y ) is a nilpotent CW complex. Moreover, if e : Y → Y0 is the rationalization of Y, then the induced map e : Mf (X, Y ) →
Me◦f (X, Y0 ) obtained by composition with e, is the rationalization of Mf (X, Y ). The
analogous result holds on based mapping spaces.
Proof. When X is a finite complex, Milnor [13] shows that Mf (X, Y ) has the homotopy type of a (countable) complex. The result on the nilpotence and the rationalization of Mf (X, Y ) in this case is [10, Theorem 2.5]. In Case b the fact that
Mf (X, Y ) has the homotopy type of a complex is due to P. Kahn [12]. For nilpotence, we observe that the proof of [11, Theorem A] goes through if Y is a finite
Postnikov piece (as opposed to being homologically finite). Similarly, the result
on the rationalization of Mf (X, Y ) in Case b can be deduced from [11, Theorem
B].
In view of the proposition, we state our results in terms of simply connected
complexes with finitely many, finitely generated homotopy groups. We remark,
however, that we could work equally well with simply connected finite complexes
X with finite-dimensional rational homotopy (i.e.
concern Theorem 3)
P
n≥2 dimQ πn (X)
⊗ Q < ∞ ).
Minimal Models. We next establish notation for working with differential graded
algebras (DGAs) and Sullivan’s theory of minimal models. Our references for rational homotopy theory are [2, 4, 18, 19].
By a DGA, (A∗ , d), we mean a connected commutative graded algebra A∗ over
Q together with an algebra derivation d of degree one whose square is zero. Given
a rational vector space V, we denote by Λn(V ) the free graded algebra generated
in degree n > 0 by the elements of V . Alternately, if x1, · · · , xl are a basis for V
6
SAMUEL SMITH
we may write Λn (x1, · · · , xl ) for Λn (V ). A graded algebra B ∗ is free if B∗ may be
written in the form B ∗ =
N∞
n=1 Λn (Vn ).
In this case, we will write Vect n(B∗ ) = Vn
for the vector space of degree n generators of B ∗ .
A DGA (B∗ , d) is minimal if B∗ is free as graded algebra, B ∗ =
N∞
n=1 Λn (Vn ),
where the graded vector space ⊕∞
n=1 Vn admits a well-ordered, homogeneous basis
xα such that dxα is a polynomial without linear term in the xβ with β < α. In
particular, if B∗ is free and simply connected then (B ∗ , d) is minimal if and only if
the image of d is contained in the decomposables of B ∗ .
If (A∗ , d) is a DGA and B ∗ is a free graded algebra given as above, then we may
form a new DGA (C ∗ , d0), which we denote by C ∗ = A∗ ⊗d0 B∗ , where, as graded
algebra, C ∗ = A∗ ⊗ B∗ , and where the differential d0 is defined by d0 |A∗ = d, while
d0 on B∗ is some degree one derivation from B ∗ to the cocycles of (A∗ , d). Note that
defining d0 is equivalent to defining linear maps d0 : Vn → H n+1 (A∗ , d).
If X is a nilpotent CW complex, then we have the de Rham-Sullivan DGA
(Ω∗ (X), δX ) which has the property that H(Ω∗ (X), δX ) ∼
= H ∗ (X, Q) by an isomorphism which is induced by a chain map to the rational cochains of X. We
say (A∗ (X), dX ) is a model for X if there is a chain map from (A∗ (X), dX ) to the
de Rham-Sullivan DGA which induces an isomorphism on cohomology. A model
(A∗ (X), dX ) is a minimal model for X if (A∗ (X), dX ) is a minimal DGA. Sullivan
[18] proves the existence and uniqueness (up to DGA isomorphism) of a minimal
model for X as hypothesized. He shows, moreover, that the minimal model of X
is, in fact, a unique invariant of the rational homotopy type of X. We will write
(M∗ (X), dX ) for the minimal model of X.
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
7
3. Results of Thom and a Calculation. We begin this section by recalling
results of Thom in [20]. Let G be any abelian group and n ≥ 1. Then, for any
complex X, Thom observes that the components of M (X, K(G, n)) represent a
single homotopy type. Since K(G, n) has the structure of abelian topological group
so does M0 (X, K(G, n)). Therefore, M0 (X, K(G, n)) has the homotopy type of a
product of Eilenberg-Maclane spaces. Thom constructs an isomorphism
(3.1)
πp (M0 (X, K(G, n))) ∼
= H n−p(X, G),
which is defined as follows. Given α ∈ πp (M0 (X, K(G, n))), by the exponential
law α is represented by a map F : S p × X → K(G, n) with F|∗×X = ∗. Let
ιn ∈ H n (K(G, n), G) be the fundamental class. Then, since F|∗×X = ∗, we may
write F ∗(ιn ) = sp ⊗ an−p ∈ H n (S p × X, G) for some an−p ∈ H n−p (X, G) where
sp ∈ H p (S p , Z) is a fixed generator. The isomorphism sends α to an−p.
Given a fibration q : E → B and any map f : X → E, we have a fibration
q : Mf (X, E) → Mq◦f (X, B), where q is the map defined by composition with
q. Thom argues that if q is a principal fibration with fibre F = K(G, n), then a
component of the fibre of q may be identified as M0 (X, K(G, n)). Thus for principal
q
fibrations K(G, n) ,→ E → B we have a fibre sequence on free mapping spaces of
the form
q
M0 (X, K(G, n)) → Mf (X, E) → Mq◦f (X, B).
Corresponding to this fibre sequence we have a long exact sequence on homotopy
of the form
∂
· · · → πp (Mf (X, E)) → πp (Mq◦f (X, B)) → πp−1 (M0 (X, K(G, n))) → · · · .
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SAMUEL SMITH
As regards the boundary homomorphism ∂ in this sequence, Thom gives the following characterization which takes into account the identification of πp−1(M0 (X, K(G, n)))
with H n+1−p (X, G).
Let kn+1 ∈ H n+1(B, G) be the k-invariant for the principal fibration K(G, n) ,→
q
E → B. Then, given F : S p × X → B with F|∗×X = q ◦ f representing α ∈
πp (Mq◦f (X, B)), we have that F ∗(kn+1 ) ∈ H n+1 (S p × X, G) and, as such, we may
write
F ∗(kn+1 ) = sp ⊗ an+1−p + 1 ⊗ an+1 .
However, since F|∗×X admits a lifting to E we see that (F|∗×X )∗ (kn+1 ) = 0 so that
actually
F ∗ (kn+1) = sp ⊗ an+1−p ∈ H n+1 (S p × X, G),
for some an+1−p ∈ H n+1−p (X, G). Thom argues that the image of α ∈ πp(Mq◦f (X, B))
under the boundary homomorphism
∂ : πp (Mq◦f (X, B) → πp−1 (M0 (X, K(G, n))) ∼
= H n+1−p(X, G)
identified in H n+1−p (X, G) is the cohomology class an+1−p . We use these results
to prove
Lemma 3.2. Let X and Y be simply connected complexes with finitely many,
finitely generated homotopy groups. Let Vn = πn(Y ) ⊗ Q, n > 1. Choose N such
that Vn = 0 for n > N. Then, the rational homotopy groups of the free mapping
space M0 (X, Y ) are given, for p > 0, by
πp (M0 (X, Y )) ⊗ Q ∼
=
N
M
H n−p(X, Vn ).
n=2
Proof. Let Yn, n = 2, . . . , N, be the terms in the Postnikov decomposition of the
rationalization Y0 of Y. Then for each n = 3, . . . , N, we have a principal fibration
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
9
pn : Yn → Yn−1 with fibre K(Vn , n), and we may assume that YN = Y0. By the
above, we have fibre sequences
pn
M0 (X, K(Vn , n)) → M0 (X, Yn ) −→ M0 (X, Yn−1).
Our method will be to show that, for each such fibre sequence, the boundary homomorphism
∂ : πp (M0 (X, Yn−1)) → πp−1(M0 (X, K(Vn , n)))
is zero for all p > 0.
Before proving this fact let us observe that it implies the result. By Proposition 2.1 and our assumptions on X and Y, we have, regarding the rationalization
(M0 (X, Y ))0 of M0 (X, Y ), that
(M0 (X, Y ))0 ' M0 (X, Y0) = M0 (X, YN ).
If all boundary homomorphisms vanish, as will be shown, then, from the exact
homotopy sequences corresponding to the above fibrations, we see inductively that
πp (M0 (X, Y )) ⊗ Q = πp(M0 (X, YN ))
∼
= πp(M0 (X, YN −1 )) ⊕ πp(M0 (X, K(VN , N )))
∼
= πp(M0 (X, YN −2 )) ⊕ πp(M0 (X, K(VN −1 , N − 1))) ⊕ πp (M0 (X, K(VN , N )))
∼
= ··· ∼
=
N
M
πp (M0 (X, K(Vn , n))).
n=2
To recover the statement of the lemma, we use Thom’s result (3.1) that
πp (M0 (X, K(Vn , n))) ∼
= H n−p (X, Vn).
To show the boundary homomorphism
∂ : πp (M0 (X, Yn−1)) → πp−1(M0 (X, K(Vn , n)))
is zero we use the characterization given it by Thom, recalled above.
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SAMUEL SMITH
Let α ∈ πp (M0 (X, Yn−1)). Choose F : S p × X → Yn−1 with F|∗×X = ∗ representing α. Let kn+1 ∈ H n+1 (Yn−1, Vn ) be the k-invariant of the principal fibration
pn
K(Vn , n) ,→ Yn → Yn−1. Then we have that ∂(α) ∈ πp−1(M0 (X, K(Vn , n))) is
identified as an+1−p ∈ H n+1−p (X, Vn) where
F ∗ (kn+1) = sp ⊗ an+1−p ∈ H n+1 (S p × X, Vn).
To show ∂(α) = 0 then it suffices to show F ∗ : H n+1 (Yn−1, Q) → H n+1 (S p × X, Q)
is zero. We prove the latter fact by considering the DGA map
M∗ (F ) : (M∗ (Yn−1 ), dYn−1 ) → (M∗ (S p × X), dS p ×X )
induced on minimal models by F : S p × X → Yn−1.
By the Künneth formula for minimal models [19, page 75] the minimal model of
S p × X is given as
(M∗ (S p × X), dS p ×X ) ∼
= (M∗ (S p ) ⊗ M∗ (X), dS p ⊗ dX ).
Recall that (M∗ (S p ), dS p ) is given by
(
Λp (xp ),
dS p = 0,
∗
p
(M (S ), dS p ) =
Λp (xp ) ⊗dSp Λ2p−1(y2p−1), dS p (y2p−1 ) = x2p
for p odd,
for p even.
Thus for p odd or even, M∗ (S p × X) decomposes as graded vector space in the
form
(3.3)
M∗ (S p × X) = 1 ⊗ M∗ (X)
M
xp ⊗ M∗ (X)
M M
i6=0,p
Mi(S p ) ⊗ M∗(X) ,
where we write Mi (S p ) for the elements of degree i in M∗ (S p ). Put
A∗ =
L
i6=0,p M
i
(S p ) ⊗ M∗ (X), and observe that A∗ is, in fact, a sub-DGA (albeit
without unit) of M∗ (S p × X). We denote the restriction of dS p ×X to A∗ by dA.
Since Yn−1 is simply connected and n-coconnected (i.e. πq (Yn−1 ) = 0 for q ≥ n
) by the Sullivan isomorphism [2, page 65], the minimal model for Yn−1 is generated
by
elements
of
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
11
degree < n. Thus Mn+1(Yn−1 ) is additively generated by decomposable elements.
Let
Q
zn+1 ∈ Mn+1(Yn−1) be of the form zn+1 =
0 < ri < n. Then
M∗ (F )(zn+1 ) = M∗ (F )
i zi
where zi ∈ Mri (Yn−1 ) and
Y Y
zi =
M∗ (F )(zi ) ∈ Mn+1 (S p × X).
i
i
Using (3.3) we may write M∗ (F )(zi ) in the form
M∗ (F )(zi ) = 1 ⊗ ai + xp ⊗ bi + Ci
where ai ∈ Mri (X), bi ∈ Mri −p (X), and Ci ∈ Ari . But now, since F|∗×X = ∗,
we see that each ai = 0. We conclude that actually
M∗ (F )(zn+1) =
Y
i
M∗ (F )(zi ) =
Y
i
(xp ⊗ bi + Ci) ∈ An+1 .
Since Mn+1 (Yn−1) is additively generated by such decomposables zn+1 , we have
shown that M∗ (F ) maps Mn+1(Yn−1 ) into An+1 .
Now, since H i (S p , Q) = 0 for i 6= 0, p, the DGA (A∗ , dA) is, by its definition,
acyclic; i.e. H ∗ (A∗ , dA) = 0. As a result of the preceding paragraph then, the map
induced by M∗ (F ) on degree n + 1 cohomology is zero. Finally, recall that the
map induced by M∗ (F ) on degree n + 1 cohomology is just F ∗ : H n+1 (Yn−1, Q) →
H n+1(S p × X, Q), and the proof is complete.
Remark. Lemma 3.2 asserts that the differentials in the Federer spectral se2
quence [6] Eij
= H −i (X, πj (Y )) ⇒ πi+j (M0 (X, Y )) have finite order. The referee
has pointed out that this fact is well-known and easy to prove when X and Y are
spectra and the spectral sequence is the Atiyah-Hirzebruch spectral sequence.
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SAMUEL SMITH
4. Proof of Theorem 1. The proof of Theorem 1 consists in comparing the
rational homotopy groups of M0 (X, X) with those of M1 (X, X). In preparation for
the proof, we follow Thom’s approach to obtain a characterization (Lemma 4.4)
of the boundary homomorphisms in the long exact homotopy sequence of certain
mapping space fibrations.
For l = 2, . . . , n+1, let Vl be a rational vector space and let Xn =
pn+1
Qn
l=2
K(Vl , l).
Let K(Vn+1 , n + 1) ,→ Xn+1 −→ Xn be a principal fibration with classifying map
kn+2 : Xn → K(Vn+1 , n + 2). Given a space X and any map fn+1 : X → Xn+1 let
fn = pn+1 ◦fn+1 : X → Xn . We seek to characterize the boundary homomorphisms
∂ : πp (Mfn (X, Xn )) → πp−1(M0 (X, K(Vn+1 , n + 1)))
(4.1)
of the long exact sequence on homotopy corresponding to the fibre sequence
M0 (X, K(Vn+1 , n + 1)) → Mfn+1 (X, Xn+1 ) → Mfn (X, Xn ).
Since Xn is a product of Eilenberg-Maclane spaces, from Thom’s result (3.1) we
have
(4.2)
πp (Mfn (X, Xn )) ∼
=
n
O
l=2
H l−p (X, Vl ) ∼
=
n
O
l=2
H l−p (X, Q) ⊗ Vl∗ ,
where Vl∗ = Hom(Vl , Q). Similarly, πp−1 (M0 (X, K(Vn+1 , n+1))) ∼
= H n−p+2(X, Q)⊗
∗
Vn+1
. Using these identifications, we view ∂ in (4.1) as a map
(4.3)
∂:
n
O
l=2
Let dl = dimQ Vl∗ ,
∗
H l−p (X, Q) ⊗ Vl∗ → H n−p+2(X, Q) ⊗ Vn+1
.
l = 2, . . . , n + 1, and let xil ,
i = 1, . . . , dl , be a ba-
∗
∗
sis for Vl∗ . Now kn+2 induces a linear map kn+2
: Vn+1
→ H n+2 (Xn , Q) on ra-
tional cohomology. Observe that H ∗ (Xn , Q) is the free graded algebra given by
H ∗ (Xn , Q) =
Nn
l=2
Λl (Vl∗ ). Thus the elements of H n+2 (Xn , Q) may be viewed as
∗
polynomials in the variables xil , 2 ≤ l ≤ n. Let pj = kn+2
(xj,n+1) ∈ H n+2(Xn , Q)
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
13
be the corresponding polynomials in xil , 2 ≤ l ≤ n. Then, regarding ∂ in (4.3), we
have
Lemma 4.4. Given al0 −p ∈ H l0 −p (X, Q) with 2 ≤ l0 ≤ n,
dn+1
∂(al0 −p ⊗ xi0 l0 ) =
X
j=1
al0 −p ·
fn∗
∂pj
∂xi0 l0
∗
⊗ xj,n+1 ∈ H n−p+2(X, Q) ⊗ Vn+1
,
for 1 ≤ i0 ≤ dl0 .
Proof. Let pl : Xn → K(Vl , l) be the projection onto the l th factor of Xn , l = 2, . . . n.
Define F : S p × X → Xn by requiring that
(
1 ⊗ xi0 l0 + sp ⊗ al0 −p ∈ H l0 (S p × X, Q)
∗
(pl ◦ F ) (xil ) =
1 ⊗ xil
∈ H l (S p × X, Q)
for i = i0 , l = l0
otherwise,
where sp ∈ H p (S p , Z) is a fixed generator, and xil = fn∗ (xil ) ∈ H l (X, Q). We claim
that F represents an element α ∈ πp (Mfn (X, Xn )). To see this, let i2 : X → S p ×X
be the inclusion i2 (x) = (∗, x). Observe that for each xil ,
(F ◦ i2 )∗ (xil ) = i∗2 (F ∗ (xil )) = i∗2 (1 ⊗ xil + sp ⊗ al0 −p ) = xil = fn∗ (xil ),
where is either one or zero according to whether or not (i, l) = (i0 , l0). It follows
that F|∗×X ' fn : X → Xn , which implies the claim. It is easy to check that this
α ∈ πp (Mfn (X, Xn )) corresponds to al0 −p ⊗ xi0l0 ∈ H l0 −p (X, Q) ⊗ Vl∗0 under the
identification (4.2).
According to Thom’s characterization of the boundary homomorphism (§3), to
prove the lemma we must show that for 1 ≤ j ≤ dn+1,
∗
F ◦
∗
kn+2
(xj,n+1)
= sp ⊗ al0 −p ·
fn∗
∂pj
∂xi0 l0
∈ H n+2 (X, Q),
∗
where we recall that pj = kn+2
(xj,n+1) ∈ H n+2(X, Q) is a polynomial in the
xil , 2 ≤ l ≤ n.
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SAMUEL SMITH
Write pj in the form
pj = (xi0 l0 )n1 m1 + · · · + (xi0 l0 )nt mt + p0j ,
where p0j is a polynomial and the mk are monomials which do not involve the
variable xi0l0 . Using the fact that s2p = 0, we compute
∗
F ∗ ◦ kn+2
(xj,n+1) = F ∗ (pj )
= F ∗ (xi0 l0 )n1 m1 + · · · + (xi0 l0 )nt mt + p0j
= 1 ⊗ xi0 l0 + sp ⊗ al0 −p
n1
+ 1 ⊗ xi0 l0 + sp ⊗ al0 −p
(1 ⊗ fn∗ (m1 )) + · · ·
nt
(1 ⊗ fn∗ (mt )) + 1 ⊗ fn∗ (p0j )
= n1 sp ⊗ al0 −p (xi0 l0 )n1−1 fn∗ (m1 ) + · · · + nt sp ⊗ al0 −p (xi0 l0 )nt −1fn∗ (mt )
+ 1 ⊗ (xi0 l0 )n1 fn∗ (m1 ) + · · · + 1 ⊗ (xi0l0 )nt fn∗ (mt ) + 1 ⊗ fn∗ (p0j )
∂pj
∗
= sp ⊗ al0 −p · fn
+ 1 ⊗ fn∗ (pj )
∂xi0 l0
∂pj
∗
.
= sp ⊗ al0 −p · fn
∂xi0 l0
(Note : fn∗ (pj ) = (kn+2 ◦ fn )∗ (xj,n+1) = 0 since kn+2 ◦ fn ' ∗.)
We are now prepared to prove
Theorem 1. Let X be a simply connected CW complex with finitely many, finitely
generated homotopy groups. Then, the components of M (X, X) are all of the same
rational homotopy type if and only if all the k-invariants of X are of finite order.
Proof. If the k-invariants of X are all of finite order, then the rationalization X0
of X is a product of (rational) Eilenberg-Maclane spaces – a finite product since
X has finite-dimensional rational homotopy. Thus X0 has the homotopy type of
an abelian topological group and so the components of M (X, X0 ) are of the same
homotopy type. Sufficiency now follows easily from the fact that, by Proposition
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
15
2.1, given any f : X → X the map
e : Mf (X, X) → Me◦f (X, X0 )
is the rationalization of Mf (X, X), where e : X → X0 is the rationalization of X.
To prove necessity we show that if such a complex X has an infinite order kinvariant, then the rational homotopy groups of M1 (X, X) are not isomorphic to
those of M0 (X, X).
Suppose X is as in the hypothesis and has an infinite order k-invariant. Let
{Xn }∞
n=2 be the spaces in the Postnikov decomposition of the rationalization X 0 of
X. Since X has finite-dimensional rational homotopy, we may suppose that XN =
X0 for some N >> 0. We are given principal fibrations pn : Xn → Xn−1 with fibre
K(Vn , n),
where
Vn
=
πn(X)
⊗
Q
and
n-equivalences fn : X0 → Xn with pn ◦ fn = fn−1 . We take fN : X0 → XN = X0
to be the identity.
Let f : X → X be any map. For each n we have the fibre sequence
pn
M0 (X, K(Vn , n)) → Mfn ◦e◦f (X, Xn ) −→ Mfn−1 ◦e◦f (X, Xn−1 ).
Thus, for each p > 0, corresponding to the above fibre sequence we have the exact
sequence
(pn )∗
πp (M0 (X, K(Vn , n))) → πp (Mfn ◦e◦f (X, Xn )) −→ πp (Mfn−1 ◦e◦f (X, Xn−1)).
Now X2 = K(V2 , 2) and
MfN ◦e◦f (X, XN ) = Me◦f (X, X0 ) ' (Mf (X, X))0 .
Since, by the above sequence,
dimQ πp (Mfn ◦e◦f (X, Xn )) ≤ dimQ πp (M0 (X, K(Vn , n)))+dimQ πp (Mfn−1 ◦e◦f (X, Xn−1)),
16
SAMUEL SMITH
and since πp (M0 (X, K(Vn , n))) ∼
= H n−p (X, Vn ), we see by an easy induction that,
for p > 0,
(4.5)
dimQ πp (Mf (X, X)) ⊗ Q ≤
N
X
dimQ H n−p (X, Vn).
n=2
In Lemma 3.2 we showed that, if f : X → X is taken to be the constant map,
then the inequality (4.5) is actually an equality for all p > 0. To prove the theorem,
we take f : X → X to be the identity and show that there exists p > 0 for which
the inequality (4.5) is strict. To show this, it suffices to prove that, for some p > 0
and some n in the range 2 ≤ n ≤ N, the boundary homomorphism
∂ : πp (Mfn ◦e (X, Xn )) → πp−1(M0 (X, K(Vn+1 , n + 1))),
corresponding to the exact sequence on homotopy of the fibration
pn+1
M0 (X, K(Vn+1 , n + 1)) → Mfn+1 ◦e (X, Xn+1) −→ Mfn ◦e (X, Xn )
(i.e the fibration in the case where f = 1X ) is nontrivial.
We are assuming that some k-invariant of X is of infinite order. It follows
from this that the corresponding k-invariant of X0 is nontrivial. Suppose kn+2 ∈
H n+2(Xn , Vn+1 ) is the first nontrivial k-invariant of X0 . Then Xn is a product of
rational Eilenberg-Maclane spaces; in fact, Xn =
Qn
l=2
K(Vl , l). With the notation
∗
∗
of Lemma 4.4, since kn+2
: Vn+1
→ H n+2 (Xn , Q) is nontrivial we may choose
∗
∗
xj0,n+1 ∈ Vn+1
such that pj0 = kn+2
(xj0,n+1 ) ∈ H n+2(Xn , Q) is a nonzero polyno-
mial in the variables xil , 2 ≤ l ≤ n. Let xi0 l0 be a variable appearing in pj0 . Then,
according to Lemma 4.4, the boundary homomorphism
∂ : πl0 (Mfn ◦e(X, Xn )) → πl0 −1(M0 (X, K(Vn+1 , n + 1)))
with identifications as in (4.3), is given on 1 ⊗ xi0 l0 ∈ H 0 (X, Q) ⊗ Vl∗0 by
dn+1
X
∂pj
∗
(fn ◦ e)
∂(1 ⊗ xi0 l0 ) =
⊗ xj,n+1 ∈ H n−l0 +2 (X, Q).
∂xi0 l0
j=1
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
Since pj0 ∈ H n+2 (Xn , Q) involves the the variable xi0 l0 , certainly
∂pj0
∂xi0 l0
17
is nonzero
in H n−l0 +2 (Xn , Q). Since fn is an n-equivalence and e is a rational equivalence,
∂p
is nonzero in H n−l0 +2 (X, Q). Thus ∂ : πl0 (Mfn ◦e (X, Xn )) →
(fn ◦ e)∗ ∂xij0l
0 0
πl0 −1 (M0 (X, K(Vn+1 , n + 1))) is nontrivial.
5. The Rational Homotopy of M1 (X, X). The space M1 (X, X), often written
G1(X), is an H-space with multiplication defined by composition of maps. Its
classifying space BG1 (X) (see [5]), is the universal cover of the classifying space,
BG(X) , for Hurewicz fibrations with fibre the homotopy type of X, as constructed
by Stasheff [17] and Allaud [1]. Determining the rational homotopy type of BG1 (X)
is an important and, in general, difficult problem. (See [18, pgs 313-14] for a
description of the Quillen minimal model of BG1 (X) and [21] for an application
of this model.) On the other hand, since M1 (X, X) is an H-space it has the
rational homotopy type of a product of rational Eilenberg-Maclane spaces. Thus
determining the rational homotopy type of M1 (X, X) is equivalent to computing
its rational homotopy groups. In [16], we use Lemma 4.4 to compute the rational
homotopy groups of M1 (X, X) (and thus, by degree shifting, the rational homotopy
groups of BG1 (X) ) for spaces X whose rational cohomology is a tensor product of
free algebras and truncated polynomial algebras – a class of spaces which includes
products of spheres, projective spaces and Eilenberg-Maclane spaces. To indicate
the method, we prove
18
SAMUEL SMITH
Theorem 2. Let X = S r1 × · · · × S rk × S t1 × · · · × S tl , where the ri are even, the
ti odd and ri , ti > 1. Then the rationalization of M1 (X, X) is given by
(M1 (X, X))0 =
k 2rY
i −1
Y
i=1 p=1
where
bip
= dimQ (H
2ri −p+1
l Y
ti
Y
i
K(Qbp , p) ×
K(H ti −p (X, Q), p) ,
i=1 p=1
(X, Q)) − dimQ (H
ri −p+1
(X, Q)).
Proof. To begin, observe that
M1 (X, X) ≈
k
Y
ri
Mpi (X, S )
i=1
!
×
l
Y
ti
!
Mqi (X, S ) ,
i=1
where pi : X → S ri and qi : X → S ti are the projections. Since ti is odd,
(S ti )0 = K(Q, ti ). It follows from Proposition 2.1 and Thom’s result (3.1) that
(Mqi (X, S ti ))0 =
ti
Y
K(H ti −p (X, Q), p).
p=1
It remains then to take n = ri to be even and compute the rational homotopy
groups of Mp (X, S n ), where p = pi : X → S n is the projection.
Let e : S n → (S n )0 be the rationalization of S n . Let π = e◦p : X → (S n )0. Then,
by Proposition 2.1, we have that (Mp (X, S n ))0 ' Mπ (X, (S n )0 ). Now the Postnikov
q
decompositon of (S n )0 is of the form K(Q, 2n − 1) ,→ (S n )0 → K(Q, n), with kinvariant
k2n = ι2n ∈ H 2n(K(Q, n), Q) where ιn ∈ H n(K(Q, n), Q) is the fundamental class.
Accordingly, we have a fibre sequence of the form
q
M0 (X, K(Q, 2n − 1)) → Mπ (X, (S n )0 ) → Mq◦π (X, K(Q, n)).
By Lemma 4.4, the boundary homomorphism
∂ : πp (Mq◦π (X, K(Q, n))) → πp−1 (M0 (X, K(Q, 2n − 1))),
identified as in (4.3), is given by
∂(an−p ⊗ ιp ) = 2an−p xn ∈ H 2n−p(X, Q),
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
19
where xn = (q ◦ π)∗ (ιn) ∈ H n (X, Q) and an−p ∈ H n−p(X, Q). Now multiplication
by xn gives a linear injection of H n−p(X, Q) into H 2n−p(X, Q). (Note that xn ∈
H n(X, Q) corresponds to a nontrivial element of H n (S n , Q) viewed in H n(X, Q).)
Thus ∂ is injective for all p > 0. Since the homotopy groups of M0 (X, K(Q, 2n −
1)) and Mq◦π (X, K(Q, n)) are given by Thom’s isomorphism (3.1) the result now
follows from a dimension count.
6. M0 (X, X)∗ as H-space. In [3], Brown gives an example of spaces X and Y
with X not a suspension and Y not an H-space such that space of based inessential
maps, M0 (X, Y )∗ , has the structure of H-space. The following lemma provides
more examples of this phenomenon:
Lemma 6.1. Let X be a simply connected complex with only two nonvanishing,
finitely generated homotopy groups, say πn(X) and πm (X), with 1 < n < m. Then
M0 (X, X)∗ =
m−1
Y
K(H m−i (X, πm (X)), i).
i=1
Proof. If Y is an H-space and X is any complex, define Φ : Y × M0 (X, Y )∗ →
M0 (X, Y ) by Φ(y, f)(x) = y · f(x), where the product is taken in Y. We claim that
Φ is a weak equivalence. To prove this, we consider the evaluation fibration,
ρ
M0 (X, Y )∗ ,→ M0 (X, Y ) → Y,
where ρ is the map which evaluates a function in M0 (X, Y ) on the basepoint of X.
It is easy to see that Φ is a fibre preserving map from the trivial fibration over Y
with fibre M0 (X, Y )∗ to the evaluation fibration. Moreover, Φ induces the identity
map on the fibre over the basepoint of Y and a homotopy equivalence on Y. Our
claim follows from the five-lemma.
20
SAMUEL SMITH
An easy consequence of this fact and Thom’s result (3.1) is that given a K(π, n)
with n ≥ 1 and π abelian and any complex X, we have a weak equivalence
(6.2)
M0 (X, K(π, n))∗ 'w
n−1
Y
K(H n−i(X, π), i).
i=1
To prove the lemma, we view X as the total space of the principal fibration
p
K(πm (X), m) ,→ X → K(πn (X), n), and observe that there then exists a fibre
sequence of based mapping spaces of the form
i
p
M0 (X, K(πm (X), m))∗ → M0(X, X)∗ → M0 (X, K(πn, n))∗ .
Since X is n − 1 connected n > 1, H q (X, πn(X)) = 0, for all 0 < q < n. By (6.2),
M0 (X, K(πn(X), n))∗ is weakly contractible and so i is a weak equivalence. By
Proposition 2.1, i is a map between CW complexes and so the result follows from
(6.2) and Whitehead’s theorem.
7. A Result of Haefliger. Up to now we have concerned ourselves primarily
with computing rational homotopy groups of function spaces, for which Thom’s
approach of using Postnikov towers is well-suited. Our proof of Theorem 3, however,
involves determining the rational homotopy type of a component of a function space.
As Thom’s work in [20] indicates, an effective approach to this problem is to analyze
the relevant evaluation maps. In this section, we recall a result of Haefliger and
apply it to deduce a characterization of certain evaluation maps.
Let X and Z be connected complexes, Vn a rational vector space, and n > 0. In
[7, §2.1], Haefliger characterizes maps of the form
ϕ : Z → M (X, K(Vn , n)),
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
21
as follows. By the exponential law, ϕ is adjoint to a map ϕ
e : X × Z → K(Vn , n).
By Thom’s result, M (X, K(Vn , n)) =
Qn
i=0
Ki where Ki = K(H n−i(X, Vn ), i). Let
ϕi : Z → Ki denote ϕ composed with the ith projection. Observe that
H i (Ki , Q) ∼
= Hn−i(X, Q) ⊗ Vn∗
where we view Hn−i(X, Q) as the dual space of H n−i(X, Q) and Vn∗ = H n(K(Vn , n), Q).
According to Haefliger, we have
Lemma 7.1. (Haefliger) Given a0 ⊗ v ∈ Hn−i(X, Q) ⊗ Vn∗ ∼
= H n−i(Ki , Q) then
ϕ∗i (a0 ⊗ v) = a0 ∩ ϕ
e∗ (v)
where a0 ∩ (a ⊗ b) = a0(a)b, for a ⊗ b ∈ H ∗ (X, Q) ⊗ H ∗ (Z, Q) ∼
= H ∗ (X × Z, Q).
We apply Haefliger’s lemma to characterize the evaluation map
ε : X × M (X, K(Vn , n)) → K(Vn , n),
defined by ε(x, f) = f(x), and its restriction to a component Mf (X, K(Vn , n)) of
M (X, K(Vn , n)) which we denote by
εf : X × Mf (X, K(Vn , n)) → K(Vn , n).
Choose an additive basis {aij |i = 0, . . . , n; j = 1, . . . , βi} of H ≤n (X, Q), where
aij ∈ H i(X, Q) and βi = dimQ (H i(X, Q)). Let bij ∈ Hi (X, Q), j = 1, . . . , βi be the
basis dual to the aij . Let a0 = a0,1 = 1 ∈ H 0 (X, Q) and let b0 = b0,1 = 10 be its
dual. Let v ∈ Vn∗ = H n(K(Vn , n), Q). Observe that
ε∗ (v) ∈ H n(X × M (X, K(Vn , n)), Q)
∼
=
∼
=
n
M
H n−i(X, Q) ⊗ H i (M (X, K(Vn , n)), Q)
i=0
H n−i(X, Q) ⊗ H i 
i=0
n
M

n
Y
j=0

Kj , Q  .
22
SAMUEL SMITH
Since Mf (X, K(Vn , n) is connected,
ε∗f (v) ∈
n
M
i=0

H n−i(X, Q) ⊗ H i 
n
Y
j=1

Kj , Q  .
Recalling that H i (Ki , Q) = H i (K(H n−i(X, Vn ), i), Q) ∼
= Hn−i(X, Q) ⊗ Vn∗ , we
prove
Lemma 7.2. (cf. [15, pgs 323-4])
i)
∗
ε (v) =
βi
n X
X
i=0 j=1
ii)
ε∗f (v) =
aij ⊗ (bij ⊗ v),
βi
n−1
XX
i=0 j=1
aij ⊗ (bij ⊗ v) + f ∗ (v) ⊗ 1.
Proof. We deduce Part i from Haefliger’s result (Lemma 7.1). Note that the adjoint
of ε
ϕ : M (X, K(Vn , n)) → M (X, K(Vn , n))
is just the identity map. Let ϕi : M (X, K(Vn , n)) → Ki be the projection. Then, by
Lemma 7.1, if bn−i,j ⊗ v ∈ Hn−i(X, Q) ⊗ Vn∗ = H i(Ki , Q),
ϕ∗i (bn−i,j ⊗ v) = bn−i,j ∩ ε∗ (v).
On the other hand, ϕ∗i (bn−i,j ⊗ v) = bn−i,j ⊗ v ∈ H i (M (X, K(Vn , n)), Q), since ϕ
is the identity. Since {aij } is an additive basis for H ≤n(X, Q) and
ε∗ (v) ∈
n
M
i=0
H n−i(X, Q) ⊗ H i (M (X, K(Vn , n)), Q),
we may write
ε∗ (v) =
βi
n X
X
i=0 j=1
an−i,j ⊗ cij
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
23
for some cij ∈ H i (M (X, K(Vn , n)), Q). But now we have
bn−i,j ⊗ v = ϕ∗i (bn−i,j ⊗ v)
= bn−i,j ∩ ε∗ (v)
= bn−i,j ∩ (an−i,j ⊗ cij )
= cij .
This proves Part i.
For ii we observe that the composition
εf
X ,→ X × Mf (X, K(Vn , n)) → K(Vn , n)
is just the map f : X → K(Vn , n), which fact, together with i, implies the result.
8. Proof of Theorem 3.
Theorem 3. Let X be a simply connected complex with finitely many, finitely
generated homotopy groups whose rationalization is a two-stage Postnikov system.
Then
M0 (X, X) 'Q X × M0 (X, X)∗ .
Proof. Our technique will be to construct the minimal model for M0 (X, X). We
follow the approach of Møller and Raussen [15].
We assume that the nontrivial rational homotopy groups of X occur in dimensions n and m with 1 < n < m. Put Vn = πn(X) ⊗ Q and Vm = πm (X) ⊗ Q. Then
the rationalization X0 of X is given by the pull-back diagram
K(Vm , m)
/ X0
P
p
K(Vn , n)
q
k
/ K(Vm , m + 1)
24
SAMUEL SMITH
where q : P → K(Vm , m+1) is the path/loop fibration and where k ∈ H m+1 (K(Vn , n), Q)
is the rational k-invariant of X. If k is zero than X0 is an H-space. By the first
paragraph of Lemma 6.1, we have that
M0 (X, X0 ) ' X0 × M0 (X, X0 )∗ .
The result now follows, in this case, from Proposition 2.1 and we may assume then
that k is nontrivial.
By [4, Lemma 3.2], the rational Postnikov tower of X determines the minimal
model, (M∗ (X), dX ), of X. Specifically, (M∗ (X), dX ) is of the form
M∗ (X) = Λn(Vn∗ ) ⊗dX Λm (Vm∗ ).
Regarding the differential dX , note that dX may be viewed as a linear map
dX : Vm∗ → H m+1 (K(Vn , n), Q) and so as an element of H m+1 (K(Vn , n), Vm ).
According to the lemma, the differential coincides with the rational k-invariant k
of X.
Since k is nontrivial, H m+1 (K(Vn , n), Q) 6= 0. It follows that n|m + 1. Since
n > 1, we cannot have that n divides m also. Given the form of (M∗ (X), dX ) we
conclude that H m (X, Q) ∼
= ker{dX : Vm∗ → H m+1 (K(Vn , n), Q)}. We assume, at
first, that dX is injective so that H m (X, Q) = 0.
The fibration p : X0 → K(Vn , n) induces a fibration on free mapping spaces
p : M0 (X, X0 ) → M0 (X, K(Vn , n)).
Since, under our current assumptions, [X, K(Vm , m)] ∼
= H m (X, Vm ) = 0, the mapping space M (X, K(Vm , m)) is connected and it follows that the the fibre of p is
M (X, K(Vm , m)). By Thom’s result, the rational cohomology of M (X, K(Vm , m))
is free as graded algebra. We wish to apply the Hirsch Lemma [4, Lemma 3.1] to the
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
25
fibration p and so obtain a model for M0 (X, X0 ). To apply the lemma, however, we
must first show that the rational cohomology of M (X, K(Vm , m)) is transgressive
(see [4, page 257]) with respect to the fibration p.
To this end, consider the fibration on mapping spaces
q : M (X, P ) → M0 (X, K(Vm , m + 1)),
induced by q : P → K(Vm , m + 1), with fibre M (X, K(Vm , m)). From the exponential law and the definition of P it is clear that M (X, P ) is homeomorphic to the path
space on M0 (X, K(Vm , m+1)). Thus q is just the path/loop fibration over a product
of Eilenberg-Maclane spaces. As such, the rational cohomology of M (X, K(Vm , m))
is easily seen to be transgressive with respect to q. To show that it is transgressive
with respect to p, we show that the fibration p : M0 (X, X0 ) → M0 (X, K(Vn , n)) is
the pullback of q via the map
k : M0 (X, K(Vn , n)) → M0 (X, K(Vm , m + 1)).
Let E denote the total space of the pullback of q by k. Then it is easy to check
that
E = {f : X → X0 | p ◦ f ' ∗}.
Thus M0 (X, X0 ) is a full component of E. To show equality, we consider the
exact sequence of sets of homotopy classes of maps corresponding to the fibration
p
K(Vm , m) ,→ X0 → K(Vn , n), a portion of which is given as
p∗
[X, K(Vm , m)] → [X, X0] → [X, K(Vn, n)].
Note that, by our assumption, [X, K(Vm , m)] = H m (X, Vm ) = 0. Thus p∗ is injective as a map of sets and it follows that E = M0 (X, X0).
26
SAMUEL SMITH
We can now apply the Hirsch Lemma to obtain a model for M0 (X, X0 ). By
Thom’s result, M (X, K(Vm , m)) and M0(X, K(Vn , n)) are products of EilenbergMaclane spaces. Thus the minimal models of these spaces are simply their rational
cohomology algebras with trivial differentials. By the Hirsch Lemma, a model
(A∗ (M0 (X, X0 )), δ) for M0 (X, X0 ) is given in the form
A∗ (M0 (X, X0 )) = H ∗ (M0 (X, K(Vn , n)), Q) ⊗δ H ∗ (M (X, K(Vm , m)), Q)
where the differential δ is just the transgression of the fibration p. We seek to
determine the differential explicitly.
Write M0 (X, K(Vm , m + 1)) =
Qm+1
i=1
Ki where Ki = K(H m−i+1 (X, Vm ), i). Let
ki : M0 (X, K(Vn , n)) → Ki
denote k composed with the projection. Since p is the pull-back of q by k, the k i
are the k-invariants for M0 (X, X0 ), and, as such, they determine the transgression
of p.
To be more explicit, write
∗
H (M (X, K(Vm , m)), Q) =
m
Y
Λi(Hm−i (X, Vm ))
i=1
where we identify Hm−i (X, Vm ) = H m−i (K(H m−i (X, Vm ), i), Q). Also, let us write
Hm−i (X, Vm ) = Hm−i(X, Q) ⊗ Vm∗ . Then given
bm−i,j ⊗ v ∈ Hm−i(X, Q) ⊗ Vm∗ = Vecti (H ∗ (M (X, K(Vm , m)), Q)),
we note that bm−i,j ⊗ v ∈ Hm−i (X, Vm ) = H i+1 (K(H m−i (X, Vm ), i + 1), Q) =
H i+1 (Ki+1 , Q). Thus k∗i+1 (bm−i,j ⊗ v) ∈ H i+1(M0 (X, K(Vn , n)), Q). By the Hirsch
Lemma, the differential
δ : Vecti (H ∗ (M (X, K(Vm , m)), Q)) → H i+1 (M0 (X, K(Vn , n)), Q)
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
27
is given by the equation
(8.1)
δ(bm−i,j ⊗ v) = k∗i+1 (bm−i,j ⊗ v).
We now apply Haefliger’s result (Lemma 7.1). Note that k is adjoint to the map
g = k ◦ ε0 where ε0 : X × M0 (X, K(Vn , n)) → K(Vn , n) is the evaluation map
restricted to the component of the constant map in M (X, K(Vn , n)). By (8.1) and
Lemma 7.1,
(8.2)
δ(bm−i,j ⊗ v) = k∗i+1 (bm−i,j ⊗ v)
= bm−i,j ∩ g∗ (v)
= bm−i,j ∩ ε∗0 (k∗ (v)),
Let x1 , . . . , xs be a basis for H n(K(Vn , n), Q). Let v1 , . . . , vt be a basis for Vm∗ .
Recall k∗ = dX : Vm∗ → H m+1 (K(Vn , n), Q), where dX is the differential for the
minimal model of X. As usual, write k ∗ (vl ) = pl (x1, . . . , xs) where pl is some
homogeneous polynomial in s variables of appropriate degree. Now H j (X, Q) = 0
for 0 < j < n. By Lemma 7.2 then, the evaluation map ε0 : X ×M0 (X, K(Vn , n)) →
K(Vn , n) is given simply by the equation
ε∗0 (xi) = 1 ⊗ b0 ⊗ xi,
where we recall b0 ∈ H0(X, Q) is the dual vector to 1 ∈ H 0 (X, Q). We see then
that, for vl ∈ Vm∗ , we have
(8.3)
ε∗0 (k∗ (vl )) = ε∗0 (pl (x1, . . . , xs))
= pl (1 ⊗ b0 ⊗ x1, . . . , 1 ⊗ b0 ⊗ xs )
= 1 ⊗ pl (b0 ⊗ x1, . . . , b0 ⊗ xs).
28
SAMUEL SMITH
Combining (8.2) and (8.3), we obtain the following description of the differential
δ. Given bm−i,j ⊗ vl ∈ Vecti (H ∗ (M (X, K(Vm , m)))) for 0 < i < m we have
δ(bm−i,j ⊗ vl ) = bm−i,j ∩ ε∗0 (k∗ (vl ))
= bm−i,j ∩ (1 ⊗ pl (b0 ⊗ x1, . . . , b0 ⊗ xs))
= bm−i,j (1) · pl (b0 ⊗ x1 , . . . , b0 ⊗ xs)
= 0,
since bm−i,j (1) = 0 for 0 < i < m. Similarly, on the basis {b0 ⊗ v1 , . . . , b0 ⊗ vt} of
Vectm (H ∗ (M (X, K(Vm , m)))) we see that δ is given by
δ(b0 ⊗ vl ) = pl (b0 ⊗ x1 , . . . , b0 ⊗ xs ).
We conclude that the model (A∗ (M0 (X, X0 )), δ) may be written in the form
m−1
O
Λi (Hm−i(X, Vm )) ,
A∗ (M0 (X, X0 )) = Λn(Vn∗ ) ⊗dX Λm (Vm∗ ) ⊗
i=1
where dX is just the differential of the minimal model of X.
It follows that
A∗ (M0 (X, X0 )) is, in fact, the minimal model for M0 (X, X0 ). Since the minimal
model of X is a factor in the minimal model of M0 (X, X0 ), by uniqueness, we
obtain
M0(X, X0 ) 'Q X ×
m−1
Y
K(H m−i (X, Vm ), i).
i=1
Now, using the proof of Lemma 6.1 and Proposition 2.1, it is easy to see that
(M0 (X, X)∗ )0 =
Qm−1
i=1
K(H m−i (X, Vm ), i). Thus, since M0 (X, X) 'Q M0 (X, X0 ),
the theorem is proved in the case where dX : Vm∗ → H m+1 (K(Vn , n), Q) is injective.
For the general case, we write Vm∗ = W ∗ ⊕ V ∗ where V ∗ = ker{dX : Vm∗ →
H m+1 (K(Vn , n), Q)}. Then we may decompose X0 as the product X0 = Y0 ×
K(V, m) where Y0 is a rational two-stage Postnikov system with injective k-invariant.
Let Y be a two-stage Postnikov system with finitely generated homotopy groups
RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS
29
whose rationalization is Y0 . Then, by the above, we have that M0 (Y, Y ) 'Q Y ×
M0 (Y, Y )∗ .
Using Proposition 2.1 and an induction on the height of a Postnikov tower, it is
easy to show that, given a simply connected complex Z with finitely many, finitely
generated homotopy groups, there is a weak equivalence
(M0 (Z, Z))0 'w M0(Z0 , Z0),
(8.4)
where (M0 (Z, Z))0 is the rationalization of M0 (Z, Z). The analogous result holds for
the space of based inessential maps. Applying (8.4) to our rational decomposition
of M0 (Y, Y ) we get
M0(Y0 , Y0) 'w Y0 × M0 (Y0 , Y0)∗
(8.5)
From (8.4), (8.5) and the exponential law we obtain the following chain of equivalences
(M0 (X, X))0 'w M0(Y0 × K(V, m), Y0 × K(V, m))
(8.6)
≈ M0 (Y0 × K(V, m), Y0 ) × M0 (Y0 × K(V, m), K(V, m))
≈ M0 (K(V, m), M0 (Y0 , Y0)) × M0 (X0 , K(V, m))
'w M0(K(V, m), Y0 × M0 (Y0 , Y0)∗ ) × M0 (X0 , K(V, m))
≈ M0 (K(V, m), Y0 ) × M0 (K(V, m), M0 (Y0 , Y0)∗ ) × M0 (X0 , K(V, m)).
Now, since Y0 is m + 1 coconnected and K(V, m) is m − 1-connected, we see
that, for p > 0,
πp(M0 (K(V, m), Y0 )∗ ) ∼
= [K(Vm , m), Ωp (Y0 )] = 0, so that
M0 (K(V, m), Y0 )∗ is weakly contractible. From the evaluation fibration (see Lemma
6.1)
ρ
M0 (K(V, m), Y0 )∗ ,→ M0 (K(V, m), Y0 ) → Y0
30
SAMUEL SMITH
we obtain that
M0 (K(V, m), Y0 ) 'w Y0 .
(8.7)
From Lemma 6.1 and (8.4), we have
M0 (Y0 , Y0)∗ 'w
m−1
Y
K(H m−i (Y0 , W ), i),
i=1
so that, by Thom’s result (3.1),
M0 (K(V, m), M0 (Y0 , Y0)∗ ) 'w
(8.8)
m−1
Y
K(H m−i (Y0 , W ), i).
i=1
Also, we have
M0(X0 , K(V, m)) 'w
(8.9)
m
Y
K(H m−i (X, V ), i).
i=1
Substituting lines (8.7), (8.8) and (8.9) into (8.6), we obtain
M0 (X, X) 'Q Y0 ×
m−1
Y
i=1
K(H m−i (Y0 , W ), i) ×
= Y0 × K(V, m) ×
'Q X ×
m−1
Y
m−1
Y
i=1
m
Y
K(H m−i (X, V ), i)
i=1
K(H m−i (Y0 , W ) ⊕ H m−i (X, V ), i)
K(H m−i (X, Vm ), i)
i=1
'Q X × M0 (X, X)∗ ,
and the proof is complete.
Remark. Hansen [9, Corollary 3.5] proves that for n ≥ 8,
n 6≡ 15 mod 16,
n 6= 11, 27 and
M0 (S n , S n ) is not homotopy equivalent to S n × M0 (S n , S n)∗ .
Thus there is no analogue of Theorem 3 in ordinary homotopy theory.
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school of mathematics, university of minnesota, minneapolis, minnesota
55455