On weak homotopy equivalences
between mapping spaces
Carles Casacuberta and Jose L. Rodriguez*
Abstract
Let Sn+ denote the n-sphere with a disjoint basepoint. We give condi-
tions ensuring that a map h: X ! Y that induces bijections of homotopy
classes of maps [Sn+; X] ~= [Sn+; Y ] for all n 0 is a weak homotopy equ*
*iv-
alence. For this to hold, it is sufficient that the fundamental groups of
all path-connected components of X and Y be inverse limits of nilpotent
groups. This condition is fulfilled by any map between based mapping spac*
*es
h: map*(B; W ) ! map *(A; V ) if A and B are connected CW-complexes. The
assumption that A and B be connected can be dropped if W = V and the
map h is induced by a map A ! B.
From the latter fact we infer that, for each map f, the class of f-loc*
*al
spaces is precisely the class of spaces orthogonal to f and f ^ Sn+for n *
* 1
in the based homotopy category. This has useful implications in the theory
of homotopical localization.
________________________________
*The authors were partially supported by DGICYT grant PB94-0725
1
0 Introduction
It is well known that unbased homotopy classes of maps from spheres are not suf*
*fi-
cient to recognize weak homotopy equivalences in general; see Section 1 for det*
*ails
about this claim. Thus, there is no unbased analogue of the Whitehead theorem,
stating that a map h: X ! Y between connected CW-complexes that induces bijec-
tions of based homotopy classes of maps [Sn; X] ~= [Sn; Y ] for all n is a homo*
*topy
equivalence [14, Theorem V.3.5].
In fact, there is no set of spaces Kffsuch that maps between CW-complexes
inducing bijections of unbased homotopy classes of maps from Kfffor all ff are
necessarily homotopy equivalences. This was proved by Heller in [11, Corollary *
*2.3].
(Of course, any family of representatives of all homotopy types of CW-complexes
suffices to recognize homotopy equivalences, but this is a proper class, not a *
*set.)
On the other hand, in the homotopy theory praxis it is frequent to encounter
situations where one would like to prove that certain maps between function spa*
*ces
are homotopy equivalences; see e.g. [2], [9]. This can be an arduous task, sin*
*ce
function spaces usually fail to be path-connected and their components can be of
distinct homotopy types. The results in this article aim to simplify this task *
*when-
ever possible.
We denote by [A; X] the set of based homotopy classes of maps from A to X,
and by A+ the union of A with a disjoint basepoint. Thus, [Sn+; X] is identifie*
*d with
the set of unbased homotopy classes of maps from the n-sphere Sn to X. All spac*
*es,
including function spaces, are endowed with the compactly generated topology.
Suppose that a map h: X ! Y induces bijections [Sn+; X] ~=[Sn+; Y ] for all*
* n. In
Section 1 we prove that such a map h is a weak homotopy equivalence if and only
if the induced homomorphism of fundamental groups, h*: ss1(X; x) ! ss1(Y; h(x)),
is surjective for every choice of a point x 2 X. This condition is fulfilled i*
*n many
important cases, namely
o if ss1(Y; y) is nilpotent for all y (see Theorem 1.8 below), or also
o if ss1(X; x) and ss1(Y; y) are both HZ -local for all x and y (see Theore*
*m 4.2).
The reader is referred to [4] and [5] for a discussion of HZ-local groups.
2
From these observations we infer the following general result about maps be*
*tween
function spaces.
Theorem 0.1 Let h: map *(B; X) ! map *(A; Y ) be any map of based function
spaces, where A and B are connected CW-complexes. Then the following statements
are equivalent:
(1) h is an integral homology equivalence.
(2) h is a weak homotopy equivalence.
(3) h induces bijections [Sn+; map *(B; X)] ~=[Sn+; map *(A; Y )] for n 0.
Of course, this is not true if we remove the assumption that A and B be con-
nected, as every space X is homeomorphic to map *(S0; X). However, using other
methods, we prove the following.
Theorem 0.2 Let f: A ! B be any map between (not necessarily connected) CW-
complexes, and let h: map *(B; X) ! map *(A; X) be induced by f, where X is any
space. Then h is a weak homotopy equivalence if and only if it induces bijectio*
*ns
[Sn+; map *(B; X)] ~=[Sn+; map *(A; X)] for n 0. (0.*
*1)
In view of these results, it is tempting to believe that a map h: X ! Y ind*
*ucing
bijections [Sn+; X] ~=[Sn+; Y ] for all n is necessarily an integral homology e*
*quivalence.
We show that this is not the case, by exhibiting a counterexample in Section 1.
Our main motivation for embarking in this study was Dror Farjoun's approach*
* to
homotopical localization [9], [10]. For a map f: A ! B of CW-complexes, a space*
* X
is called f-local [9] if the map of function spaces map *(B; X) ! map *(A; X) i*
*nduced
by f is a weak homotopy equivalence. Thus, Theorem 0.2 asserts precisely that
unbased homotopy classes of maps from spheres suffice to recognize f-local spac*
*es.
Moreover, note that (0.1) can also be written as
[B ^ Sn+; X] ~=[A ^ Sn+; X] for n 0. (0.*
*2)
3
The fact that (0.2) characterizes f-local spaces is very useful in certain cons*
*tructions
of homotopy idempotent functors. Indeed, the results contained in a preliminary
version of this article have been exploited in [10, p. 14].
Similarly, if A is any CW-complex, then a map g: X ! Y is said to be an
A-equivalence if the arrow map *(A; X) ! map *(A; Y ) induced by g is a weak ho-
motopy equivalence [3], [10, Section 2.A]. From Theorem 0.1 it follows that unb*
*ased
homotopy classes of maps from spheres suffice again to characterize A-equivalen*
*ces,
provided that the space A is connected. This is useful, for instance, in the co*
*ntext
of [10, p. 54].
Acknowledgements We thank E. Dror Farjoun for providing stimulating ideas
and for his encouragement, that we appreciate.
1 Unbased homotopy classes of maps
We keep denoting by X+ the union of a space X with a disjoint basepoint. Recall
from [14, Section III.1] that, if a space Y is path-connected, then for each sp*
*ace X
the set [X+ ; Y ] of unbased homotopy classes of maps from X to Y can be identi*
*fied
with the set of orbits of [X; Y ] under the usual action of ss1(Y ). In particu*
*lar, [S1+; Y ]
corresponds bijectively with the set of conjugacy classes of elements in ss1(Y *
*).
A map h: X ! Y between topological spaces is a weak homotopy equivalence if
it induces a bijection of path-connected components ss0(X) ~=ss0(Y ) together w*
*ith
isomorphisms
ssn(X; x) ~=ssn(Y; h(x)) for n 1 and every x 2 X : (1.*
*1)
Even though it might seem plausible, condition (1.1) cannot be replaced by the
condition that h induces bijections
[Sn+; X] ~=[Sn+; Y ] for n 1 : (1.*
*2)
The following source of counterexamples is extracted from [11, x 2].
Example 1.1 Let N be a torsion-free group such that any two nontrivial eleme*
*nts
are conjugate; embed N into a larger group G with a single nontrivial conjugacy
4
class of elements as well (this can be done by iterating suitable HNN construct*
*ions;
see [13, Theorem 6.4.6]). Then the induced map of classifying spaces h: BN ! BG
induces bijections [Sn+; BN] ~=[Sn+; BG] for all n. However, h is not a weak ho*
*motopy
equivalence, as it fails to be surjective on the fundamental group.
Constructions of this kind also serve to discard the belief that a map h: X*
* ! Y
inducing bijections [Sn+; X] ~=[Sn+; Y ] for n 0 is an integral homology equiv*
*alence.
Here is a counterexample.
Example 1.2 Let G be the union of an ascending chain of groups
N = N0 N1 N2 : : :
where, for each i 0, the group N2ihas precisely one nontrivial conjugacy class*
* of
elements and N2i+1 is acyclic. Then the inclusion of N into G induces bijectio*
*ns
[Sn+; BN] ~=[Sn+; BG] for all n. Yet, BG is acyclic and BN need not be.
We next give conditions under which (1.2) suffices to guarantee that h is a*
* weak
homotopy equivalence.
Lemma 1.3 Let G be any group and let ': A ! B be a ZG-module homomorphism
inducing a bijection of orbits. Then ' is an isomorphism.
Proof. If '(a) = 0 = '(0), then a is in the orbit of 0 and hence a = 0. This sh*
*ows
that ' is a monomorphism. Moreover, for every b 2 B we may write b = x . '(a) =
'(x . a) for some a 2 A and x 2 G, showing that ' is an epimorphism. ]
Theorem 1.4 Suppose that a map h: X ! Y induces bijections [Sn+; X] ~=[Sn+;*
* Y ]
for n 0. Then h is a weak homotopy equivalence if and only if the induced homo-
morphism of fundamental groups is surjective on each path-connected component.
Proof. One implication is obvious. To prove the converse, we may assume without
loss of generality that X and Y are path-connected. By assumption, the homo-
morphism h*: ss1(X) ! ss1(Y ) induces a bijection of conjugacy classes. Then h**
* is a
5
monomorphism, since h*(x) = 1 = h*(1) forces x = 1; hence, our additional assum*
*p-
tion guarantees that h* is in fact an isomorphism. Now, for each n 2, we have
a homomorphism h*: ssn(X) ! ssn(Y ) of ZG-modules, where G = ss1(X) ~= ss1(Y ),
and each of these is bijective on orbits. It then follows from Lemma 1.3 that h
induces isomorphisms of all homotopy groups. ]
We denote the lower central series of a group G by 0G = G and iG =
[G; i-1G] for i 1. The proof of the next result is an exercise on commutator
calculus and induction, that we omit.
Lemma 1.5 Suppose that a group homomorphism ': G ! K is surjective on con-
jugacy classes. Then, for every i 1, each element y 2 K can be written as
y = fli'(i), where fli 2 iK and i = ji-1i-1 with ji-1 2 i-1G. ]
As an immediate consequence, we have
Proposition 1.6 If a group homomorphism ' : G ! N is surjective on conjugacy
classes and N is nilpotent, then ' is an epimorphism. ]
We also record the following variation, which will be used later.
Proposition 1.7 Suppose given a commutative diagram
ffi ffi-1 ff1
G : : :! Gi+1 ! Gi ! : : :! G2 ! G1
' # # 'i+1 # 'i # '2 # '1
K : : :! Ki+1 ! Ki ! : : :! K2 ! K1;
fii fii-1 fi1
where ' is induced by passage to the inverse limit. If all Gi and Ki are nilpo*
*tent
and ' is surjective on conjugacy classes, then ' is an epimorphism.
Proof. By refining the inverse systems if necessary, we may assume that Gi and *
*Ki
have nilpotency class less than or equal to i. Take any element y 2 K, and deno*
*te
it by (y1; y2; y3; : :):, with yi 2 Ki, and fii-1(yi) = yi-1. We are going to c*
*onstruct an
element x 2 G such that '(x) = y. By Lemma 1.5, we can write y = fl1'(1) with
fl1 2 1K. Set x1 = (1)1 2 G1. Then '1(x1) = y1, since 1K1 is trivial. Next, wri*
*te
6
y = fl2'(2) with fl2 2 2K, 2 = j11, j1 2 1G. Set x2 = (2)2. Then '2(x2) = y2
and, moreover, ff1(x2) = x1, since 1G1 is trivial. By continuing the same way, *
*we
obtain an element x = (x1; x2; x3; : :):2 G such that 'i(xi) = yi for all i, so*
* that
'(x) = y. ]
Note that Propositions 1.6 and 1.7 can also be proved by resorting to Lemma*
* 4.1
below, since every inverse limit of nilpotent groups is HZ-local.
From Theorem 1.4 and Proposition 1.6 we infer the main result of this secti*
*on:
Theorem 1.8 Let h: X ! Y induce bijections [Sn+; X] ~=[Sn+; Y ] for n 0. Su*
*ppose
that the fundamental group of each path-connected component of Y is nilpotent. *
*Then
h is a weak homotopy equivalence. ]
2 Maps between function spaces
Given topological spaces B and X with basepoint, we denote by map *(B; X) the
space of all based maps from B to X with the compactly generated topology. The
space map *(B+ ; X) of unbased maps is denoted, as usual, by map (B; X). For a
based map g: B ! X, we denote by map *(B; X)g the path-connected component
containing g, and similarly for unbased maps.
We recall from [14, Theorem I.7.8] that, if B is well pointed, then for any*
* X the
following sequence is a fibre sequence, where the second arrow is evaluation at*
* the
basepoint:
map *(B; X) ! map (B; X) ! X: (2.*
*1)
In fact, for every map g: B ! X we have a fibre sequence
[
map *(B; X)j ! map (B; X)g ! X; (2.*
*2)
j
where j ranges over a set of representatives of based homotopy classes of maps *
*such
that j ' g by an unbased homotopy.
We shall exploit the crucial fact, explained in [12, Theorem II.2.5], that *
*if A
is a connected CW-complex of finite dimension, then for every space X the path-
connected components of map *(A; X) are nilpotent. In view of Theorem 1.8, this
7
remark proves Theorem 0.2 in the special case when A is finite-dimensional and
connected. In the rest of this section we prove Theorem 0.2 in its full general*
*ity.
Proof of Theorem 0.2:
Suppose given a map f: A ! B of CW-complexes, not necessarily connected. Let X
be an arbitrary space, and assume that the induced map
h: map *(B; X) ! map *(A; X)
gives rise to a bijection [B; X] ~=[A; X] together with bijections
[Sn+; map *(B; X)] ~=[Sn+; map *(A; X)] for n 1 :
We can write
Y
map *(B; X) = map *(B0; X) x map (Bb; X);
b
where B0 denotes the basepoint component and a point b has been chosen in each
of the other connected components of B; indeed, we denote by Bb the connected
component containing b. The same notation is used with A.
We start by showing that there is no restriction in assuming that f induces*
* a
bijection of connected components ss0(A) ~= ss0(B). Firstly, suppose that B ha*
*s a
component Bb which does not intersect f(A). Then the condition [B; X] ~=[A; X]
forces [(Bb)+ ; X] to be trivial. In addition, the exponential law yields
[B; map *(Sn+; X)] ~=[A; map *(Sn+; X)] (2.*
*3)
for all n 1. Hence, for each n 1, the set [(Bb)+ ; map *(Sn+; X)] has a sing*
*le
element, and this implies that ssnmap (Bb; X) is zero, since all its elements l*
*ie in a
single orbit under the action of the fundamental group. It follows that map (Bb*
*; X)
is weakly contractible. Secondly, suppose that two components Aa, Ac map into
the same component Bb. Then (2.3) forces map *(Sn+; X) to be path-connected for
n 1. This implies that [Sn+; X] is trivial for all n; therefore, ssn(X) is als*
*o trivial
for all n, and X is weakly contractible.
8
We therefore assume that f induces a bijection ss0(A) ~= ss0(B). Then h det*
*er-
mines a collection of maps
map *(B0; X0) ! map *(A0; X0)
(2.*
*4)
map (Bf(a); Xx) ! map (Aa; Xx);
and we are led to showing that each of these is a weak homotopy equivalence. By
Theorem 1.4, it suffices to prove that the induced homomorphisms of fundamental
groups are surjective. For simplicity of notation, we shall assume from now on *
*that
A, B and X are path-connected, and drop most subscripts. Using (2.2), for each
choice of a based map g: B ! X we obtain a commutative diagram with exact rows
'1 '2 '3
ss2(X) ! ss1map *(B; X)g ! ss1map (B; X)g ! ss1(X)
# = # '7 # '8 # =
ss2(X) ! ss1map *(A; X)gf ! ss1map (A; X)gf ! ss1(X):
'4 '5 '6
Lemma 2.1 Suppose given a commutative diagram where the rows are exact,
M >! G !! Q
# '0 # ' # '
N >! K !! R:
Then the following hold:
(a) If ' is surjective on conjugacy classes, so is '.
(b) If N im ' and ' is an epimorphism, then ' is an epimorphism.
(c) If N is nilpotent, ' is an epimorphism, and ' is surjective on conjugacy
classes, then ' is an epimorphism.
(d) If ' is an isomorphism and ' is surjective on conjugacy classes, then '0*
* is
surjective on conjugacy classes.
(e) If N im ' and ' is surjective on conjugacy classes, then ' is surjective*
* on
conjugacy classes.
9
Proof. Parts (a) and (b) are straightforward. In order to prove (c), we show *
*that
N im ' and apply (b). Thus, pick any element y 2 N. By assumption, we may
write y = z'(u)z-1 with z 2 K and u 2 G. Then z = '(v) for some v 2 G, and
y1 = z'(v)-1 belongs to N. Hence, if we set x0 = vuv-1, then we have
y = y1'(vuv-1)y-11= [y1; '(x0)]'(x0);
where both y1 and '(x0) belong to N. By arguing as in Lemma 1.5, we find that
y = fli'(i) for all i 1, with fli 2 iN, which finishes the argument.
We next prove (d). As in the previous part, start with an element y 2 N and
write it as y = y1'(vuv-1)y-11with y1 2 N and u; v 2 G. Now the injectivity of '
ensures that vuv-1 2 M, as required. Part (e) is straightforward. ]
In our situation, the assumption that f induces a bijection [B; X] ~= [A; *
*X]
guarantees that the arrow im '3 ! im '6 is an isomorphism. Since '8 is surjecti*
*ve
on conjugacy classes, the restriction
im '2 ! im '5
is surjective on conjugacy classes, by part (d) of the above lemma. Furthermor*
*e,
the commutative diagram
im '1 >! ss1map *(B; X)g !! im'2
# # '7 #
im '4 >! ss1map *(A; X)gf !! im '5;
in view of part (e) of the above lemma, shows that '7 is surjective on conjugacy
classes. This argument reduces our problem to the case of based mapping spaces.
Denote by Ai the ith skeleton of A, and similarly for B. Assuming that f is*
* a
cellular map, there is a commutative diagram with exact rows,
lim1ss2map *(Bi; X)g >! ss1map *(B; X)g !! lim ss1map *(Bi; X)g
# '7# #
lim1ss2map *(Ai; X)gf >! ss1map *(A; X)gf!! limss1map *(Ai; X)gf:
For every i, the spaces map *(Bi; X)g and map *(Ai; X)gf are nilpotent. By assu*
*mp-
tion, '7 is surjective on conjugacy classes. Thus, parts (a) and (c) of Lemma *
*2.1,
10
together with Proposition 1.7, imply that '7 is in fact surjective, hence compl*
*eting
the proof of Theorem 0.2. ]
3 Characterizing local spaces
The half-smash product Xx|Y of two spaces is a standard notation for X^Y+ (cf. *
*[10,
Section 2.D]). For a map f: A ! B, a space X is called f-local if map *(f; X) i*
*s a
weak homotopy equivalence [9]. Since [Sn+; map *(A; X)] ~=[Ax|Sn; X], Theorem 0*
*.2
can be reformulated as follows. This answers in the affirmative a question pos*
*ed
in [6, p. 15].
Corollary 3.1 Let f: A ! B be any map between CW-complexes. Then a space X
is f-local if and only if f induces a bijection [B; X] ~=[A; X] together with b*
*ijections
[Bx|Sn; X] ~=[Ax|Sn; X] for n 1. ]
In a more categorical language, this result implies the following. If C is*
* any
category, we say that an object X and a morphism f: A ! B are orthogonal, as
in [1] or [8], if the map of sets C(B; X) ! C(A; X) induced by f is bijective. *
*A class
of objects D is called a small-orthogonality class [1, Section 1.C] if there is*
* a set
of morphisms fffsuch that D is precisely the class of objects orthogonal to all*
* fff.
Thus, Corollary 3.1 yields:
Corollary 3.2 For each map f between CW-complexes, the class of f-local spaces
is a small-orthogonality class in the based homotopy category. ]
Indeed, a space X is f-local if and only if it is orthogonal to the set consist*
*ing of
f and fx|Sn for n 1. This remark sheds light on Dror Farjoun's argument in [9]
or [10, Section 1.B], where it is shown that the class of f-local spaces is ref*
*lective in
the based homotopy category for every map f, i.e., that f-localization exists f*
*or all
spaces.
11
4 Homology equivalences of function spaces
The possibility of the following improvement of our previous results was sugges*
*ted
by Dror Farjoun. The reader is referred to [4] for the definition and properti*
*es of
HZ-localization, i.e., localization with respect to ordinary integral homology.*
* Recall
that a space L is HZ-local if every integral homology equivalence W ! V of CW-
complexes induces a bijection of based homotopy classes of maps [V; L] ~= [W; L*
*],
and a group G is HZ-local if and only if it is isomorphic to the fundamental gr*
*oup
of an HZ-local space.
Lemma 4.1 If a group homomorphism ': G ! K between HZ-local groups is sur-
jective on conjugacy classes, then it is an epimorphism.
Proof. The assumption that ' is surjective on conjugacy classes implies that t*
*he
induced homomorphism of abelianizations, '*: H1(G) ! H1(K), is surjective. Ac-
cording to [5, Corollary 2.13], a homomorphism between HZ-local groups which
becomes surjective after abelianizing is itself surjective. ]
In view of Theorem 1.4, we have
Theorem 4.2 Suppose that a map h: X ! Y induces bijections [Sn+; X] ~=[Sn+;*
* Y ]
for n 0. If the fundamental groups of all path-connected components of X and Y
are HZ-local, then h is a weak homotopy equivalence. ]
We can now prove Theorem 0.1 as a corollary.
Proof of Theorem 0.1:
Let h: map *(B; X) ! map *(A; Y ) be any map between function spaces, where A a*
*nd
B are now assumed to be connected (and this is essential). As before, denote by*
* Ai
the ith skeleton of A and similarly for B. The space map *(A; Y ) is weakly equ*
*ivalent
to the inverse limit of the spaces map *(Ai; Y ), under the inclusions Ai ,! Ai*
*+1. Since
each space map *(Ai; Y ) is a disjoint union of nilpotent spaces for i 1, it f*
*ollows
from [4, x 12] that the space map *(A; Y ) is HZ-local. Of course, we can argue
the same way with map *(B; X). If h is an integral homology equivalence, then,
12
since its source and target are HZ-local spaces, h is a weak homotopy equivalen*
*ce.
The converse implication is well known, as it is also the fact that a weak homo*
*topy
equivalence induces bijections of unbased homotopy classes of maps from all sph*
*eres.
To prove the converse of the latter claim in our case, observe that the fundame*
*ntal
group of each path-component of map *(A; Y ) or map *(B; X) is an HZ-local grou*
*p,
so that Theorem 4.2 applies. ]
Recall that a map g: X ! Y is said to be an A-equivalence if map *(A; g) i*
*s a
weak homotopy equivalence; cf. [3], [10, Section 2.A]. As a corollary of Theore*
*m 0.1,
we obtain the following.
Corollary 4.3 Let A be any connected CW-complex. Then a map g: X ! Y is
an A-equivalence if and only if it induces a bijection [A; X] ~= [A; Y ] togeth*
*er with
bijections [Ax|Sn; X] ~=[Ax|Sn; Y ] for n 1. ]
Since [Ax|Sn; X] ~=[A; map (Sn; X)], the latter condition can of course be *
*refor-
mulated in terms of iterated free loops of g.
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vol. 61, Springer-Verlag, New York Heidelberg Berlin, 1978.
Departament de Matematiques, Universitat Autonoma de Barcelona,
E-08193 Bellaterra, Spain, e-mail: casac@mat.uab.es, jlrodri@mat.uab.es
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