Miller Spaces and Spherical Resolvability of
Finite Complexes
Jeffrey Strom
Abstract
We show that if K is a nilpotent finite complex, then K can be built
from spheres using fibrations and homotopy (inverse) limits. This is ap-
plied to show that if map *(X, Sn) is weakly contractible for all n, then
map *( X, K) is weakly contractible for any nilpotent finite complex K .
AMS Classification numbers Primary:
Secondary:
Keywords: Miller Spaces, Spherical Resolvability, Resolving Class, Closed
Class, Homotopy Limit
Discussion of Results
A Miller space is a CW complex X with the property that the space of
pointed maps from X to K , written map *(X, K), is weakly contractible
for every nilpotent finite complex K . They are named for Haynes Miller,
who proved in [12 ] that the spaces B Z =p are all Miller spaces; in fact,
he proved that map *(B Z =p, K) is weakly contractible for every finite
dimensional CW complex K .
In the stable category, one can define a Miller spectrum by requiring that
the mapping spectrum F (X, K) ' * for every finite spectum K . Since
cofibrations and fibrations are the same in the stable category, a finite
spectrum K with m cells is the fiber in a fibration K -! L -! Sn in
which L has only m - 1 cells; in the terminology of [5, 11], this means
that K is spherically resolvable with weight m. An easy induction shows
that X is a Miller spectrum if and only if F (X, Sn) ' * for every n.
1
Our goal is to prove the following unstable analog of this observation: if
map *(X, Sn) is weakly contractible for all n, then X is a Miller space.
The proof of the stable version is not available to us because cofibrations
are not fibrations, unstably. To prove our result, it is necessary to deter-
mine the extent to which a finite complex can be constructed from spheres
in a more general way, i.e., by arbitrary homotopy (inverse) limits [3] and
by extensions by fibrations.
To be more precise, we require some new terminology. We call a nonempty
class R of spaces a resolving class if it is closed under weak equivalences
and pointed homotopy (inverse) limits (all spaces and homotopy limits will
be pointed). It is a strong resolving class if it is further closed under
extensions by fibrations, i.e., if whenever F -! E -! B is a fibration with
F, B 2 R, then E 2 R.
Resolving classes are dual to closed classes as defined in [4] and [6, p. 45].
Notice that every resolving class R contains * (cf. [6, p. 47]). From
this, it follows that if F -! E -! B is a fibration with E, B 2 R, then
F 2 R. Similarly, if Aff2 R for each ff then the categorical product
ffAff2 R also. The weak product effAffis the homotopy colimit of the
finite subproducts; if for each i only finitely many of the groups ßi(Aff)
are nonzero, then the weak product has the same weak homotopy type as
the categorical product.
Let_S be the smallest resolving class that contains Sn for each n, and let
S be the smallest strong resolving class that contains_Sn_for each n. We
say that a space K is spherically resolvable if kK 2 S for some k.
This concept is related to, but not the same as, the notion of spherical
resolvability described in [5, 11]. We list some other important examples
of (strong) resolving classes below.
Examples
(a) If f : A -! B is any map then the class of all f -local spaces is a
resolving class [6, p. 5]. This includes, for example, the class of all
spaces with ßi(X) = 0 for i > n, or all h*-local spaces, where h* is
a homology theory.
(b) If P is a set of primes, then the class of all P -local spaces is a strong
resolving class.
2
(c) If f : W -! *, then the class of all f -local spaces is a strong resolv-
ing class [6, p. 5]. This includes, for example the class {K+ }, where
K+ denotes the Quillen plus construction on K [6, p. 27].
(d) More generally, if F is a covariant functor that commutes with ho-
motopy limits (and fibrations) and R is a (strong) resolving class,
then the class {K | F (K) 2 R} is also a (strong) resolving class.
This applies, for example to the functor F (K) = map *(X, K).
(e) The class {K | K is weakly contractible} is a strong resolving class.
Our proofs will proceed by induction on a certain kind of cone length [1].
Let F denote the collection of all finite type wedges of spheres. The F -
cone length clF(K) of a space K is the least integer n for which there
are cofibrations Si- ! Ki- ! Ki+1, 0 i < n, with K0 ' *, Kn ' K
and each Si 2 F . If no such n exists, then clF(K) = 1. Clearly every
finite complex K has clF(K) < 1.
With these preliminaries in place, we can state our main result.
Theorem 1 If K is a nilpotent space with clF(K) = n < 1, then
__
(a) K 2 S , and
(b) nK 2 S .
In particular, every nilpotent finite complex K is spherically resolvable in
our sense.
Our application to Miller spaces follows from the following more general
consequence of Theorem 1.
Theorem 2 Let R be a strong resolving class, and assume that X has
the property that map *(X, Sn) 2 R for each n. Then map *( X, K) 2 R
for each nilpotent space K with clF(K) < 1.
This result has many corollaries; we list a few.
Corollary 3 Let X be a space and P a set of primes.
3
(a) If map *(X, Sn) is weakly contractible for all n, then map *( X, K)
is weakly contractible for all nilpotent K with clF(K) < 1. In other
words, X is a Miller space.
(b) If map *(X, Sn) is P -local for all n, then map *( X, K) is P -local
for all nilpotent K with clF(K) < 1.
We end by making the surprising observation that a (non-nilpotent, of
course) finite complex can be a Miller space!
Example Let A be a connected 2-dimensional acyclic finite complex.
(The classifying space of the Graham-Higman group [7] is such a space; so
is the space obtained by removing a point from a non-simply connected
3-dimensional Poincar'e sphere). Since ß1(A) is equal to its commuta-
tor subgroup, there are no nontrivial homomorphisms from ß1(A) to any
nilpotent group. It follows that if f : A -! K with K a nilpotentWfinite
complex, then ß1(f) = 0 and so f factors through q : A -! A=A1 ' S2.
Since [A, S2] ~=H2(A) = 0, we conclude f ' *. Thus A is a Miller space.
One of the key points in the proof of Theorem 1 is an explicit description of
the homotopy fiber of a map B -! B [CA, which we state as Proposition
5. In an appendix, we use this result to give simple proofs of the Blakers-
Massey excision theorem [2] and the James splitting of X [10 ].
I would like to thank Robert Bruner and Charles McGibbon for suggesting
I think about Miller spaces. This work owes much to McGibbon in partic-
ular - Theorem 3(a) was conjectured in joint work with him. Thanks to
Bill Dwyer for directing me to the result of [9], which is the key to Propo-
sition 4; thanks are also due to Daniel Tanr'e for bringing Proposition 5 to
my attention.
1 Proof of Theorem 1
We begin with two supporting results.
Proposition 4 Let K be a connected nilpotent space, let R be a resolv-
ing class and let F be a functor that preserves fibrationsWand homotopy
(inverse) limits over diagrams indexed on n and L. If F ( mi=1 K) 2 R
for each m, then F (K) 2 R.
4
Proof This follows from a result of Hopkins [9, p. 222], which says that
K is homotopy equivalent to the homotopy (inverse) limit of a tower
A0- A1- . .-. An- . . .
of spaces, each of whichWis a homotopy (inverse) limit over n of a diagram
of spaces of the form mi=1 K . 2
W m
It follows that to show that kK 2 R itWsuffices to show that k( i=1 K) 2
R for all m; more generally, if F ( k( mi=1 K)) 2 R for all m then
F ( kK) 2 R.
Proposition 5 Let A -! B -! C be a cofibration, and let F be the ho-
motopy fiber of B -! C . Then
F ' A _ ( A ^ C).
Proof Convert each of the maps A -*! C , B -! C and C -=! C to a
fibrations. The total spaces and fibers form the commutative diagram
A x |CJ ______________//FA
| JJJ | AAA
| JJJ | AAA
| JJ%% | A__
| C ______|______//*
| | |
| | | |
| | | |
fflffl| || fflffl| |
A _________________//JB |
JJJ | @@@ |
JJJ | @@ |
JJJ | @@ |
JJ$fflffl|$ _fflffl|_
* ______________//C,
in which the bottom square is a homotopy pushout. A result of V. Puppe
[13 ] shows that the top square is also a homotopy pushout. Hence, the
cofiber F of the map F -! * has the same homotopy type as the cofiber
of A x C -! C , namely A _ ( A ^ C), as can be seen from the
diagram
A x C _______// C _____________//_ F
| | ||
| pushout | ||
fflffl|* fflffl| ||
A ________//A * C_____// A _ ( A ^ C).
2
5
Proof of Theorem 1 Notice that the assumption on X implies that X
is connected; we may therefore assume that K is also connected.
We prove both assertions in parallel, by induction on clF(K). If clF(K) =
1 then K is homotopyWequivalent to a connected finite type wedge of
spheres. Then each mi=1 K is a simply-connected finite typeWwedge of
spheres, and the Hilton-Milnor theorem [8, 14] shows that ( mi=1 K) is
homotopy equivalent to the weak product e Snff2 S . Since the wedge
is of finite type and simply connected, all but finitely many factors are
not i-connected for each i, so the weak product has the same homotopy
type as the categorical product. Proposition 4 proves both assertions in
the initial case.
Now assume that both statements are known for all nilpotent spaces with
F -cone length less than n, and that K is nilpotentWwith clF(K) = n.
ByWProposition 4,_ it is enough to show that n( mi=1 2K) 2 S and
( mi=1 2K) 2 S for each m.
Wm 2 W m
Write V = i=1 K . Notice that clF( i=1K)W clF(K), and the double
suspension of an F -cone decomposition of mi=1K is an F -cone decom-
position of V . Thus we may assume that V has an F -cone decomposi-
tion Si- ! Vi- ! Vi+1, 0 i < n with each Si and Vi simply-connected.
Therefore, we have a cofibration L -! V - ! W with L simply-connected,
clF(L) < n and W a simply-connected finite type wedge of spheres.
Let F denote the homotopy fiber of V - ! W . Consider the fibration
sequences
F -! V - ! W and nV - ! nW -! n-1F.
__ __
Since kW 2 S S for all k > 0, it suffices to show that F 2 S and
that n-1F 2 S .
Now we use Proposition 5 to determine the homotopy type of F :
`` '
F ' L _ (L ^ W ) ' L ^ Snff
ff
which is a finite type wedge of suspensionsWof L. If we smash an F -cone
length decomposition of L with the space ffSnff we obtain an F -cone
length decompositionWfor F - in other words, clF( F ) < n and, more
importantly, clF( li=1 F ) < n for each l.
6
W l __ n-1 W l
By the inductive hypothesis, ( i=1 F ) 2 S and ( i=1 F ) 2 S
for each l. Since L, V and W are_each simply-connected, so is F , and
Proposition 4 implies that F 2 S and that n-1F 2 S , as desired. 2
2 Proof of Theorem 2 and Corollary 3
Proof of Theorem 2 Let M be the class of all spaces Y such that
map *(X, Y ) 2 R; we have already seen that M is a strong resolving_
class. Since Sn 2 M for each n by assumption, it follows that S
M. By Theorem 1, M contains K for every nilpotent space K with
clF(K) < 1. 2
Proof of Corollary 3 Only statement 3 needs proof. Let M be the
class of all spaces K for which map *(X, cK) is weakly contractible. By
assumption Sn 2 M for each n. Since the Sullivan completion functor
commutes with fibrations and homotopy (inverse) limits indexed on n
and L,
3 Appendix: Applying Proposition 5
This section is essentially an advertisement for Proposition 5. Note that
the proof of Proposition 5 uses only elementary results on homotopy
pushouts and pullbacks. Nevertheless, it has many very important conse-
quences, among which are elementary proofs of the Blakers-Massey exci-
sion theorem [2] and the James splitting of X [10 ].
Theorem(Blakers-Massey) Let A -! B -! C be a cofibration of simply-
connected spaces with A (a - 1)-connected and C (c - 1)-connected. If
F is the homotopy fiber of B -! C then the natural map A -! F is an
(a + c - 2)-equivalence.
Proof The assumptions clearly imply that F is simply-connected. The
result follows from the fact that the map
A -! F ' A _ ( A ^ C),
7
is an (a + c - 1)-equivalence. 2
We use the notation X(n) for the n-fold smash product of X with itself.
W (n)
Theorem(James) If X is a connected space, then X ' n X .
Proof We apply Proposition 5 to the cofibration X -! * - ! X . The
homotopy fiber of * -! X is X , so we conclude that
X ' X _ (X ^ ( X))
' X _ ( X ^ X) _ (X ^ X ^ ( X))) ' . ...
The result follows by induction, using the fact that the X(n) is at least
(n - 1)-connected. 2
Finally,Wwe observe that our proof of James' theorem reveals that the
space ( nX(n)) is a wedge summand of X , even when X is not
connected.
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