DRAFT-LOOPSPACES AND FINITENESS
CLARENCE W. WILKERSON
Abstract. This expository note began as comments on a shorter
note of F. R. Cohen [5]. Cohen's paper is an elegant application of
powerful recent results in unstable homotopy theory to a problem
of interest to analysts.
Theorem : (F. R. Cohen,[5]) Let X be a simply connected finite
complex which is not contractible and let j0X be the component
of the constant map in the j-th pointed loop space of X. If j 2,
then the Lusternik-Schnirlman category of j0X is not finite.
This note includes a rederivation of the above theorem using
H-space methods of W. Browder from the 60's, [3], [4]. The aim
is to reduce the prerequisites for Cohen's theorem to those avail-
able after a second course in algebraic topology. We end with a
discussion of recent work of Lannes-Schwartz on various notions of
finiteness properties and the behavior under looping. The common
theme is extensive use of the action of the Steenrod algebra on the
cohomology of a topological space.
1. Introduction
The finiteness of the Lusternik-Schnirlman category of a space can
be thought of as one possible generalization of a finite complex. It's not
difficult to show that finiteness is most often not inherited by loopspaces
(see Theorem 1.2 below). Cohen's result shows that this fails for finite-
ness of the Lusternik-Schnirlman category as well, even if X is finite:
1.1. Theorem. [F. R. Cohen,[5]] Let X be a simply connected finite
complex which is not contractible and let j0X be the component of
the constant map in the j-th pointed loop space of X. If j 2, the
Lusternik-Schnirlman category of j0X is not finite.
A somewhat similar result has long been known - the case j = 1:
1.2. Theorem. Let X be a simply connected complex of finite type
which is not contractible.
(a) (Borel, [2]) If H*(X, k) is a finite dimensional graded vector
space for some field k, then H*( X, k) is not finite dimensional.
____________
Date: May 23, 2006.
1
2 CLARENCE W. WILKERSON
(b) If X has finite Lusternik-Schnirlman category, then X does
not have finite Lusternik-Schnirlman category.
The reader first confronted with this rather wordy introduction might
well wonder what the current author could or should add to Cohen's
three page note. The answer was originally rooted in a desire for math-
ematical economy in the prerequisites for the proof. Cohen's proof uses
results from Dwyer-Wilkerson which arise from Lannes' T-technology.
Thus on the surface at least, the proof depends on recent deep techni-
cal advances in unstable homotopy theory. These in turn are grounded
in principle in the details of the Bousfield-Kan unstable Adams spec-
tral sequence for function spaces. Most of us would have some dif-
ficulty explaining this background to our colleagues or graduate stu-
dents. Therefore, for the first half of this exposition the goal is to limit
the technology to that roughly corresponding to the earlier chapters of
Mosher-Tangora [8] - the Serre and Bockstein spectral sequences and
Steenrod operations.
This problem demonstrates the immense influence of cohomology alge-
bras and cohomology operations on the evolution during the past half
century of unstable homotopy theory. An impressive early demonstra-
tion of this power is provided by Serre's beautiful use of the precise
knowledge of the action of the Steenrod algebra on the cohomology of
Eilenberg-MacLane spaces to give asymptotic information about the
homotopy groups of finite complexes.
1.3. Theorem. (Serre, [10 ]) Let X be a finite simply connected complex
such that H*(X, Fp) 6= 0 in some positive dimension. Then there exist
infinitely many distinct positive integers {ni} for which
ssni(X) Fp 6= 0.
The following formulation of Serre's theorem is very useful:
1.4. Corollary. Let X be a simply connected finite CW complex which
is not contractible. Then j0X is not contractible for any j 0. If
X is a non-contractible simply connected finite CW complex then X is
not a finite Postnikov tower.
The work of Lannes-Schwartz [7] continues this tradition of Serre.
We sketch in the last section generalizations of 1.1 and 1.2.
1.5. Definition (Lannes-Schwartz, [7]). Let M be a graded unstable
Ap-module. If for each m 2 M, the cyclic submodule Ap(m) is finite,
then M is said to quasi-bounded or locally finite over the Steenrod
algebra Ap.
DRAFT-LOOPSPACES AND FINITENESS 3
It turns out that quasi-bounded has better properties for loopspaces
than finiteness or finite category:
1.6. Theorem (Lannes-Schwartz, [7]). Let X be a simply connected
complex such that H*(X, Fp) is non-zero in at least one positive di-
mension. Suppose further that H*(X, Fp) is quasi-bounded as a module
over the Steenrod algebra Ap. Then
a) H*( j0X, Fp) and H*( jX, Fp) are quasi-bounded for all j 0.
b) ssi(X) Fp 6= 0 for infinitely many i.
c) j0X is not contractible for any j 0.
Of course, (b) is a form of Serre's earlier result. However the Lannes-
Schwartz method generalizes to other interesting inverse connections
between the complexity of the mod p cohomology algebras and the
existence of infinite dimensional homotopy groups.
These methods imply a stronger version of 1.1:
1.7. Theorem (Section 5). Let X be a simply connected CW com-
plex such H*(X, Fp) is finite in each dimension and is quasi-bounded
over Ap. If H*(X, Fp) is non-zero in some positive dimension, then
H*( X, Fp) is not a finitely generated algebra over Fp. By iteration,
the same holds for each j0X for j > 0.
For the reader with primary interests in homotopy theory, it would
well worthwhile to followup on the results of Lannes and Schwartz,
using, for example, Schwartz's book, [9]. The non-homotopy theorist
might find that the proof of 1.1 using Browder's work is sufficient and
requires a much smaller set of prerequisites.
My interest and knowledge about this problem came from conversa-
tions with Fred Cohen at M.P.I. in Bonn in the summer of 1997. I
thank the M.P.I. for its hospitality. Fred may well regard these com-
ments as a waste of good trees.
2. Preliminaries and the proof of Theorem 1.2
We give two proofs of 1.2. The first is elementary while the second
hints at Browder's work. The eventual proof of 1.1 is a blend of these
two ideas.
2.1. Proposition. Let X be a simply connected space such that H*(X, Fp)
is of finite type over Fp. If H*( X, Fp) is finite, then H*(X, Fp) is not
finite.
4 CLARENCE W. WILKERSON
Proof of 1.2 from 2.1.Since X is a connected H-space, H*( X, Fp)
is a connected Hopf algebra. The Borel structure theorem for comm-
muative Hopf algebras of finite type over Fp [1] implies that H*( X, Fp)
as a commutative graded algebra must be a tensor product of exterior
algebras, polynomial algebras, and truncated polynomial algebras. If
X has finite cup length in cohomology, then the tensor factors of
H*( X, Fp) must be a finite number of exterior and truncated polyno-
mial algebras. That is, finite category or finite cup length imply that
H*( X, Fp) is finite.
Elementary proof of 2.1 ala Borel-Serre. Consider the Serre Spectral
Sequence of the path space fibration
X ! PX ! X.
The E2 term is H*(X, Fp) H*( X, Fp) and the differentials go to
the right and downward. This spectral sequence converges to the
cohomology of the path space PX. Since PX is contractible, the
E1 is zero except in bidegree (0, 0). If Hk(X, Fp) = 0 for k > M,
and Hj( X, Fp) = 0 for j > N then choose the M and N so that
HM (X, Fp) and HN ( X, Fp) are each nonzero. Then the E2 is concen-
trated in a rectangle from (0, 0) to (M, N). For dimensional reasons,
each element of bidegree (M, N) is an infinite cycle and the image of
the differentials from other bidegree in (M, N) is trivial. The exis-
tence of these non-zero terms in EM,N1 contradicts the contractibility
of PX.
2.2. Proposition (Second proof of 1.2). Let X be a simply connected
space with finite L.-S. category and H*(X, Fp) of finite type and nonzero
in at least one positive dimension. Then H*( X, Fp) is not finite.
Proof. Suppose that H*( X, Fp) is finite and not concentrated in de-
gree 0. Since H*( X, Z) has finite type, then the Bockstein specral
sequence implies that H*( X, Q) is a finite dimensional graded vec-
tor space. From the Borel structure theorem, the Euler characteris-
tic OH*( X, Fp) = 0. Hence OH*( X, Q) = 0. Since X is con-
nected, this implies that there is at least one odd degree N > 0 with
HN ( X, Q) 6= 0. That is, H*( X, Q) is a non-trivial exterior algebra
on odd-dimensional generators. By the Borel transgression theorem,
H*(X, Q) is thus a finitely generated polynomial algebra. This violates
the assumption of finite L.S.-category for X.
There is actually a close connection between the first proof above and
Browder's work involving Bockstein spectral sequences and Poincare
duality for H-spaces. In the Serre spectral sequence, one deals with
DRAFT-LOOPSPACES AND FINITENESS 5
differential bigraded algebras and obtains "at the corner" an automatic
infinite cycle that never bounds. In the Bockstein spectral sequence for
the homology or cohomology of a finite H-space, there is only a single
grading so there is no "corner" per se. Browder's clever use of filtrations
based on the Hopf algebra structure pushes through the basic style of
the Serre spectral sequence case, however.
3. Proof of Theorem 1 using Browder's H-space work
We again assume that the category of j0X is finite and derive a
contradiction based on finite cup length.
The basic observations in the proof of Theorem 1.2 all apply to the
proof of Theorem 1.1 . In particular, H*( j0X, Fp) is finite for each
prime p, and since it is nontrivial in positive dimensions, it is non-
trivial in positive dimensions for all primes. Moreover, H*( j0X, Z)
vanishes above some dimension d0 = dp.
The proof of Theorem 1.1 is now brutally fast - we are completely
anticipated by a 1960 theorem of W. Browder [3]:
3.1. Theorem. (Browder, 1960) If Y is an simply connected H-space
such that H*( Y, Z) is a finitely generated module over the integers Z,
then Y is homotopy equivalent to a finite product of circles.
3.2. Remark. Since the first proof of 3.1 in [Browder, [3]] uses informa-
tion at many primes, the resulting proof of 1.1 does not have a trivial
generalization to the p-local or p-complete setting. Later work of Brow-
der on homotopy commutative H-spaces does provide a suitable one
prime proof of analogues of 3.1, and thus 1.1, as does the proof in
Cohen [5]. At p = 2, Hubbuck gives a characterization of homotopy
commutative H-spaces with finite cohomology that recovers Browder's
theorem at p = 2.
In our case j0X = jX[j], where X[j] is the connective cover of X
with trivial homotopy groups in dimension j and lower. Hence B j0X =
j-1X[j] is simply connected. The remarks in the proof of Theorem
1.2 show that the groups H*( j0X, Z) are zero for large dimensions, so
the hypotheses of Browder's theorem hold and j0X h.e.S1 x . .x.S1.
Hence, contrary to Serre's theorem, ss*(X) is not non-zero in infinitely
many dimensions. Thus the assumption that the category of j0X is
finite must be false.
Sketch of Browder's proof of 3.1.: Using the finite type assumption on
H *( X, Z) yields that H*( X, Z) is p-torsion free for almost all primes
6 CLARENCE W. WILKERSON
p. Hence, for almost all primes, H*(X, Fp) is a finitely generated poly-
nomial algebra on even dimensional generators which is a commutative
Hopf algebra. Browder then uses Steenrod operations to restrict the
degree of the generators to be of the form {2pk} for p > 2. Since these
dimensions agree with those of the rational cohomology generators, and
are valid for all sufficiently large p, the only choice is that each generator
of H*(X, Fp) is in dimension 2. Since the rational rank of X is r and
that is the dimension, in fact H * ( X, Fp) has no even-dimensional in-
decomposables. This implies that H*( X, Z) is p-torsion free for each
p. Hence the composition of K(ss1( X, 1) ! X ! K(ss1( X, 1) is a
homotopy equivalence.
4. Quasi-bounded algebras
Neither finiteness or finite L.-S.-category have good inheritance prop-
erties under the operation of taking loopspaces. Lannes-Schwartz fill
this gap with the notion of quasi-boundeness as defined in the intro-
duction and show that this property is inherited under the operation
of forming pointed loopspaces.
4.1. Example. (1) If X is a finite CW complex, then X has finite
category and H*(X, Fp) is quasi-bounded for any prime p.
(2) The space Y = CP 1 has category 2 but its mod p cohomology
is not quasi-bounded for any prime p.
(3) The space S3 has quasi-bounded mod p cohomology for any
prime p, but does not have finite category since the cup length
in Fp cohomology is not bounded.
4.2. Theorem (Lannes-Schwartz [7]). Let X be a CW complex of finite
type.
(a) If X is simply connected, then H*(X, Fp) quasi-bounded implies
that H*( X, Fp) is quasi-bounded .
(b) If X is a connected H-space with universal cover Y , then if
H*(X, Fp) is quasi-bounded , then so is H*(Y, Fp).
(c) Let X be a simply connected complex of finite type such that
H*(X, Fp) is quasi-bounded. Then H*( Xj0, Fp) is quasi-bounded
for all j 0.
The next theorem is a generalization of Theorem 1.1:
4.3. Theorem. Let X be a simply connected complex of finite type such
H*(X, Fp) is quasi-bounded and Hm (X, Fp) 6= 0 for some m > 0. Then
DRAFT-LOOPSPACES AND FINITENESS 7
H*( X, Fp) is not finitely generated as an Fp-algebra. Hence X does
not have finite cup length or category. By iteration, the same holds for
each j0X.
The reader should compare Browder's similar statement in the case
that X is an H-space.
4.4. Remark. Lannes-Schwartz [7] derive many remarkable consequences
from their proof that unlike finiteness or finite category, the property
of quasi-bounded is preserved under taking loopspaces. The proof of
4.2 is quite similar to Cohen's proof of 1.1.
Proof of 4.2.Assume that H*( X, Fp) is a finitely generated algebra.
Using the Hopf algebra structure as before, we again conclude that
H*( X, Fp) is finite. Next, we appeal to the theory of p-compact
groups as developed by Dwyer-Wilkerson [6]. The p-completions of
X and X satisfy these conditions. The homotopy fixed point method
of [6] produces the existence of a non-trivial map BZ=pZ ! X if
H*( X, Fp) is non-trivial. By Lannes, such maps are detected by mod p
cohomology. We conclude that there is a non-nilpotent element of pos-
itive degree in H*(X, Fp). But this contradicts the quasi-boundedness
assumption on H*(X, Fp). Hence the assumption that H*( X, Fp) is
finitely generated as an algebra is incorrect.
There are many mapping space questions suggested by the special
case of loopspaces. For example, given CW complexes X and Y , under
what conditions is ss*(Map(X, Y )) non-zero in infinitely many dimen-
sions?
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8 CLARENCE W. WILKERSON
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Department of Mathematics, Purdue University, West Lafayette,
IN 47907
E-mail address: wilker@math.purdue.edu