Table Of ContentLecture Notes ni
Mathematics
Edited by A. Dold and B. Eckmann
763
Algebraic ygolopoT
Aarhus 1978
sgnideecorP
of a Symposium held at Aarhus,
Denmark, August 7 - ,21 1978
UNIVERSITY OF AARHUS
50TH ANNIVERSARY
11 SEPTEMBER 1978
Edited by
.J .L Dupont and .I H. Madsen
galreV-regnirpS
Berlin Heidelberg New kroY 1979
Editors
Johan Louis Dupont
Ib Henning Madsen
Matematisk Institut
Aarhus University
8000 Aarhus C
Denmark
AMS Subject Classifications (1970): 55-02, 57-02
ISBN 3-540-09721-X Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-09721-X Springer-Verlag NewYork Heidelberg Berlin
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P R E F A C E
In August 1978 a Symposium on Algebraic Topology
was held in Aarhus. The Symposium was part of the sci-
entific activity in connection with the 50th anniver-
sary of Aarhus University, and was supported by the
Danish Science Foundation and Aarhus University's Re-
search Fund.
The meeting was structured as a series of plenary
talks together with special sessions on Homotopy Theory,
Characteristic Classes and Bordism, Algebraic K- and L-
Theory, Transformation Groups and Geometry of Manifolds.
These Proceedings contain manuscripts submitted by
the invited speakers and are for the most part accounts
of the talks given.
We would like to thank everyone who contributed to
the success of the Symposium.
Aarhus, June 1979 Johan Dupont, Ib Madsen
TABLE OF CONTENTS
HOMOTOPY THEORY:
F.R. Cohen, J.C. Moore, J.A. Neisendorfer, Decomposition of loop
spaces and applications to exponents .......... I
K. Knapp, nO the bi-stable J-homomorphism ............... 13
M. Mahowald, Some homotopy classes generated by qj ........ 23
J.P. May, Applications and generalizations of the approxi-
mation theorem ................................ 38
J. McCleary, doM p decompositions of H-spaces; Another
approach ...................................... 70
CHARACTERISTIC CLASSES AND BORDISM:
W. Browder, Complete intersections and the Kervaire
invariant ..................................... 88
J.L. Dupont, Bounds for characteristic numbers of flat
bundles ....................................... 109
F. Hegenbarth, Exotic characteristic classes of spherical
fibrations .................................... 120
U. Koschorke, nO the (n+1)-tuple points of immersed n-spheres 132
L.L. Larmore, Isotopy classification of spheres in a manifold 135
A. Liulevicius, Homotopy rigidity of sturdy spaces ............ 141
B.J. Sanderson, The geometry of Mahowald orientations ......... 152
ALGEBRAIC K- AND L-THEORY:
G. Carlsson, Desuspension in the symmetric L-groups ........ 175
F.J. - B.J.Clauwens,Product formulae for surgery obstructions ..... 198
J.-Cl. Hausmann, Algebraic K-theory and flat manifolds ......... 212
E. Laitinen, I. Madsen, Topological classifications of S£2(~ p)
space forms ................................... 235
A.S. Mishchenko, C~-algebras and K-theory ...................... 262
A. Ranicki, The total surgery obstruction ................. 275
lV
M. Steinberger, On the equivalence of the two definitions of
the algebraic K-theory of a topological space 317
R.W. Thomason, First quadrant spectral sequences in algebraic
K-Theory ...................................... 332
F. Waldhausen, Algebraic K-theory of topological spaces II ... 356
TRANSFORMATION GROUPS:
S.E. Cappell, J.L. Shaneson, Pseudo-free actions I ................ 395
T. tom Dieck, Semi-linear group actions on spheres:
Dimension functions ........................... 448
S. Jackowski, T. Zukowski, P-free linear representations of P-solv-
able finite groups ............................ 458
C. Kosniowski, E. Ossa, Orientation preservin 9 involutions ........ 465
R. Lashof, Obstructions to equivariance .................. 476
A.S. Mishchenko, A.T. Fomenko, Symplectic Lie group actions ....... 504
R. Oliver, Free compact group actions on products of
spheres ....................................... 539
T. Petrie, Three theorems in transformation groups ....... 549
M. Rothenberg, Homotopy type of G spheres .................... 573
R. Schultz, Finding framed ~paCtions on exotic spheres 591
GEOMETRY OF MANIFOLDS:
D. Burghelea, The rational homotopy groups of Diff(M) and
Homeo(M) in the stability range .............. 604
S.E. Cappell, J.L. Shaneson, A counterexample on the oozing problem
for closed manifolds .......................... 627
F.T. Farrell, W.C. Hsiang, Remarks on Novikov's conjecture and the
Topological-Euclidian Space Form Problem ...... 635
M. Kreck, Isotopy classes of diffeomorphisms of (k-1)-
connected almost - parallelizable 2k-manifolds. 643
S.H. Weintraub, Inefficiently embedded surfaces in 4-manifolds. 664
L. Taylor, B. Williams, Local Surgery: Foundations and Applications 673
Decompositions of Loop Spaces and
Applications to Exponents
by
F.R. Cohen, J.C. Moore, and J.A. Neisendorfer*
We shall discuss product decompositions of loop spaces and their application
to exponents of homotopy groups. The fundamental definition is: A simply connected
space X has exponent pk at the prime p if pk annihilates the p-primary component
of ~q(X) for all .q
It is not at all obvious that there exist nontrivial simply connected finite
complexes which have exponents. But, in the 19S0's, I.M. James (for p=2 [9]) and
H. Toda (for p odd [14]) showed that the odd dimensional sphere 2n+l S has exponent
2n
p at p. Based on computational evidence, .M Barratt then conjectured that
2n+l S has the lesser exponent pn at any odd prime .p
Barratt also asked the following question [2]. Given a double suspension
for which the identity map has finite order, does it have an exponent? Barratt
obtained bounds on the exponent of the homotopy groups. However, these bounds
grew as the dimension of the homotopy group got larger. The existence of an
exponent for the simplest example, a 2-connected mod pr Moore space, remained
open.
In the 1960's, .B Gray ]7[ showed that Barratt's exponent conjecture for
2n+l S was best possible. For p odd, he constructed infinitely many elements of
order pn in E,(s2n+I). These elements stabilized mod p to elements in the image
of .J
Recently, the thesis of .P Selick [12] gave a proof of the first nontrivial
case of Barratt's conjecture. The sphere 3 S has exponent p at an odd prime .p
Studying the exponent problem for Moore spaces led us to new methods, valid
for primes greater than .3 We began by considering Samelson products in the mod p
homotopy Boekstein speezral sequence of the loops on a 2-connected mod pr Moore
r+l
space. With this machinery, we detected infinitely many elements of order p
in the homotopy groups. Further investigation of this higher torsion gave two
* The authors were supported in part by the National Science Foundation.
theorems. First, a 2-connected mod pr Moore space has exponent at most p2r+l
Second, 2n+l S has exponent pn at any prime p greater than .3
Somewhat stronger and more geometric results are true. Suppose that 2n+l S
has been localized at p greater than 3 and let l+n2lf( s2n+l)o denote the component
of the basepoint in the 2n+l fold loop space. Then the pn-th power map of
(~2n÷l s2n+l. 2n+l
Jo is null homotopic. Similarly, the p -st power map of the 4-fold
loop space of a 2-connected mod pr Moore space is null homotopic. In fact, all
exponent theorems we know reduce to null homotopic power maps in some iterated
loop space.
We get our exponent theorems as consequences of product decompositions of loop
spaces. These product decompositions are related to an old theorem of Serre [13]
and also to the Hilton-Milnor theorem [lO]. Recall Serre's theorem: if all spaces
are localized away from ,2 then ~S 2n+2 decomposes up to homotopy type into the
product 2n+l x S ~S 4n+3.
Serre proves his theorem as follows. The integral Pontrjagin ring of
~S 2n+2 is a tensor algebra on a single generator of degree 2n+l. We may interpret
a tensor algebra as a universal enveloping algebra of a free lie algebra. Serre's
decomposition comes from a decomposition of this universal enveloping algebra into
a tensor product. He constructs a geometric realization of this algebraic
decomposition by using Samelson products to construct maps.
Serre's method for decomposing loop spaces is our method also with two
differences. First, his example was simple enough that he had no need to use the
language of universal enveloping algebras. Second, we work with mod pr homology
and use Samelson products in mod pr homotopy. This means that there is a Bockstein
structure which must be compatible with our decompositions.
It is by way of this Bockstein structure that higher torsion in homotopy
influences the construction of product decompositions for loop spaces. One such
product decomposition gives a map ~2s2n+i ÷ s2n-lof spaces localized at p greater
than .3 This map has degree p on the bottom cell of ~2s2n+l and it gives a
factorization of the double loop of the degree p map on ~2s2n+l. From this
factorization, the exponent theorem for spheres follows.
.I How to prove a decomposition theorem
We shall prove a product decomposition theorem for the loops on a mod p r
Moore space where the top cell is even and p is an odd prime. The Bockstein struc-
ture is an important part of the proof. Hence, this example shows how differential
Lie algebras enter the proof of a product decomposition. But higher torsion plays
no role so that higher order Bocksteins do not enter. Higher order Bocksteins play
a role in some of the decompositions given in the next section. If we ignore Bock-
steins, the technique below can be used to prove such classical theorems as Serre's
decomposition of ~S 2n+2 localized away from 2 and the Hilton-Milnor theorem.
We will not give all details below. Those statements which are not proved
here are treated in our paper [3].
Let p be a prime and let :rp m m + S S denote the degree pr map. The Moore
space S m-I U r em' denoted pm(pr), is the cofibre of the map pr: sm-1 + sm-1.
P
Let sm{p r} denote the homotopy theoretic fibre of the map p : r S ÷ MT S "MT
Theorem. If p si an odd prime wit~ n> 0, then there is a momotopy equivalence
s2n+l{p r} x ~ V p4n+2kn+3 (pr) ÷ ~p2n+2 (pr).
k=o
Proof: Since p2n+2(pr) is the suspension ~ p2n+l(pr), the Bott-Samelson theorem
gives that H, (~p2n+2(pr); z/prz) is the tensor algebra T(u,v) with generators u
and v of degrees 2n and 2n÷l, respectively. The Bockstein B related to the short
exact coefficient sequence z/prz ÷ Z/p2rz ÷ z/prz satisfies Bv = .u This Bockstein
gives T(u,v) the structure of a differential algebra.
The tensor algebra T(u,v) si the universal enveloping algebra UL where
L=L(u,v), the free differential Lie algebra generated by u and .v
There is a natural map of differential Lie algebras L + A where A = < u,v ,>
the abelian differential Lie algebra generated by u and .v
The kernel of L ÷ A is the commutator sub-Lie algebra [L,L]. Subalgebras of
free Lie algebras are free; hence [L,L] is free. It is generated by countably many
generators, Yk = adk )uC [v,v] and k = z ad k )u( [u,v] with k > .0 That B Yk = 2Zk
follows from the derivation property of the Bockstein.
Given any short exact sequence 3 ÷ 2 L ÷ 1 L L of Lie algebras which are free
modules, UL 2 and UL 1 (~ UL 3 are isomorphic UL I modules. Hence, UL si isomorphic
to U[L,L] ~) UA as a differential U[L,L] module. This is the algebraic version of
our decomposition theorem. The proof will be completed by giving maps s2n+l{p r} ÷
~p2n+2(pr) and ~ V p4n+2kn+3(pr) ÷ ~p2n+2(pr) such that the induced maps in mod
k=o
r
p homology are isomorphisms onto U A and U[L,L], respectively. For, the loop
multiplication of ~p2n+2(pr) allows us to multiply these maps to get a map
s2n+l{p r} x ~ V p4n+2kn+3(pr) ÷ ~p2n+2(pr). The KUnneth theorem implies that this
k=o
map is a mod pr homology isomorphism. Since these spaces are (already) localized
at ,p it is a homotopy equivalence.
Extending the fibration sequence s2n+l{p 2n+1 2n+l to ÷ r} S ~ S the left gives a
principal fibration sequence ~S 2n+l + s2n+l{p r} + 2n+l S . The Serre spectral se-
quence of the latter shows that H, (s2n+l{pr}; z/prz) and H, (~s2n+l; z/prz) Q
H, (s2n+l; z/prz) are isomorphic H, (~S2n+l; z/prz) modules. The Bockstein maps
degree 2n+l isomorphically onto degree .n2
Since the composition 2n+l ~ 2n+l S + S p2n+2(pr) is null, there is a homotopy
commutative diagram in which the rows are fibration sequences up to homotopy,
GS 2n+l + s2n÷l{p r} + S 2n÷l P~ S 2n÷l
~p2n+2(pr) ~ ap2n+2 + . ÷ p2n÷2(pr)
(* denotes a point).
The map s2n÷l{p r} + ~p2n÷2(pr)is equivariant ~ith respect to the left actions
of ~S 2n÷l on s2n+l{p r} and ~p2n+2(pr). It is now easy to see that the map induces
an isomorphism of H, (s2n÷l{pr}; z/prz) onto .AU
To construct the second map, it is convenient to introduce the James construc-
tion J(X)[8]. Let X be a (pointed) space. Then J(X) is the free monoid generated
by the points of X subject to the single relation that the basepoint is the unit.
There is an injection X ÷ J(X) which is continuous in the natural topology on J(X).
Given a topological monoid G and a map X ~ ,G there is a unique extension to a mul-
tiplicative map J(X) + .G The loop space made up of loops of arbitrary length is
a topological monoid. The suspension :E X ÷ ~EX induces a multiplicative homotopy
equivalence from J(X) to ~EX for connected .X Up to homotopy, J(X) and ~EX are
interchangeable.
r
We also need some knowledge of mod p homotopy groups. If X is a space,
~m(X; z/prz) si the set of homotopy classes of maps from p(pr) to .X
If p is greater than 3 and ~X is simply connected, then ~,(~X; z/prz) si a
differential Lie algebra. The differential si the Bockstein and the Lie bracket is
the Samelson product. The Hurewicz map ~ is a morphism of differential Lie algebras
[11].
if p equals 3, the Jacobi identity for Samelson products fails. Since ew do
not use it, this proof is valid if p equals 3.
Let U and ~ be homotopy classes in n,(~p2n+2(pr); z/prD such that ~(~) = u and
~(~) = v. The iterated Samelson product adk(~) [~,~] defines a map p4n+2kn+2(pr)_+
~p2n+2(pr). Add these maps to get a map V p4n+2kn+2(pr) ÷ ~p4n+2kn+2(pr). Ex-
k=o
tend to the James construction to get a map ~ V p4n+2kn÷3(pr) + ~p2n+2(pr). The
k=o
induced map in mod pr homology is an isomorphism onto U[L,L] since [L,L] si gener-
ated by ~(adk(~)[~,~]) and their Bocksteins.
.2 Decompositions of localized loop spaces
Many decomposition theorems hold only if we localize at a prime. Accordingly,
we assume that all spaces are localized at a prime p in the remainder of this paper.
Some spaces are localized at the start, for example, mod pr Moore spaces or fibres
of degree pr maps, but, for example, we write m to S denote the result of local-
izing the m-sphere at .p
All decompositions theorem below are decompositions of localized homotopy
types. The decomposition theorems for the loop spaces of spheres, Moore spaces,
and certain related spaces provide a perspective on parts of classical homotopy
theory. Complete details are given in our papers [3,4,5,6].
Localized at an odd prime, even dimensional homotopy theory is expressible in
terms of odd dimensional homotopy theory. To be precise, we list a result of Serre
,]31[ a related result on the fibres of degree pr maps, and the decomposition
