Algebraic Topology: Proceedings, University of British Columbia, Vancouver, August 1977

280 Pages·1978·3.041 MB·English

by  Roy Douglas (auth.), Peter Hoffman, Renzo A. Piccinini, Denis Sjerve (eds.)

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
673
ciarbeglA ygolopoT
of British University Columbia, Proceedings,
,revuocnaV August 7791
Edited yb
.P Hoffman, .R Piccinini dna .D evrejS
MMIiIIIIIIIIIJ ETHICS ETH-BIB
00100000870703
galreV-regnirpS
nilreB Heidelberg New York 1978
Editors
Peter Hoffman
University of Waterloo
Department of Pure Mathematics
Waterloo, Ont. N2L 3G1
Canada
Renzo A. Piccinini
Memorial University of Newfoundland
Mathematics, Statistics and
Computer Science
St. John's, Nfld, A1B 3X7
Canada
Denis Sjerve
University of British Columbia
Department of Mathematics
Vancouver, B. C., V6T lW5
Canada
Library of Congress Catalogln| is Pubiicati,,a D|ts
Main entry under title:
Algebraic topology.
(Lecture notes in mathematies ; 675)
Proceedings of a workshop a~d conference held
July 25-Aug. 12, 1977, and sponsored by the
Canadian Mathematical Congress.
Bibliography: .p
Includes index.
.i Algebraic topology- -Congres ses. .I Hof fman,
Peter, 1941- II. Piecinini, Renzo A.,
1933- III. Sjerve~ Denis, 19~I-
IV. Canadian Mathematical Congress (Society)
.V Series: Lecture notes in mathematics
(Berlin) ; 673.
QAS.L28 no. 675 QA612 510'.8s 51~'.2
78-13254
AMS Subject Classifications (1970): 18H10, 55R20, 55D10, 55D15,
55 D99, 55 F10, 55 F15, 55 F35, 55 F50,55 G 35,55 H15,57A65
ISBN 3-540-08930-6 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-08930-6 Springer-Verlag NewYork Heidelberg Berlin
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FOREWORD
From .July 25 to August 12,1977, the Canadian Mathematical Society spon-
sored a Workshop and Conference in Algebraic Topology at the University of British
Columbia, in Vancouver.
The Workshop, which involved the active participation of Graduate Students,
consisted of a series of informal lectures devoted to the presentation of new
theories and of background material related to talks to be delivered later on;
furthermore, during this part of the meeting, there were discussions about the
research work being conducted by Graduate Students. The speakers of the Workshop
were: E.Campbell*, A.Dold, R.Douglas, S.Feder, P.Heath, P.Hoffman, R.Kane, J.
McCleary*, C.Morgan*, L.Renner* and C~Watkiss.
The lecturers of the Conference proper were: R.Body, P.Booth, A.Dold,
R.Douglas, S.Feder, H.Glover, P.Heath, P.Hilton, R.Kane, A.Liulevicius, G.Mislin,
S.Segal, L.Siebenmann, F.Sigrist, D,Sjerve, V.Snaith, J.Stasheff, U.Suter, M.
Tangora and C.Watkiss.
The articles printed in these Proceedings are based on talks given during
the Conference; we observe that not all the talks are represented here since some
speakers elected not to submit a paper.
The published papers have been divided into four areas:
A. Rational Homotopy Theory;
B. Cohomology Theories; Bundle Theory;
C. Homotopy Theory; Nilpotent Spaces; Localization;
.D Group Cohomology; Actions.
The general index has been prepared so as to make this division clear. In
each area the articles appear in alphabetical order by the name of the author or
first author.
The list of addresses of all contributors and the names of all participants
are given at the end of this volume.
.P Hoffman
R. Piccinini
D. Sjerve
*Graduate Students
CONTENTS
A. RATIONAL HOMOTOPY THEORY
.R Douglas:
The Uniqueness of Coproduct Decompositions for Algebras Over a Field ..... 1
J. Stasheff:
Rational Homotopy-Obstruction and Perturbation Theory .................... 7
.B COHOMOLOGY THEORIES: BUNDLE THEORY
A. Dold:
Geometric Cobordism and the Fixed Point Transfer ......................... 32
H. Glover, .B Homer and .G Mislin:
Immersions in Manifolds of Positive Weights .............................. 88
.R Kane:
BP Homology and Finite H-Spaces ........................................... 93
.F Sigrist and U. Suter (with the collaboration of P.J. Erard):
On Immersions CP n ~---~R 4n-2~(n) ...................................... 106
.F Sigrist and .U Suter:
On the Exponent and the Order of the Groups ~(X) ....................... I16
V. Snaith:
Stable Decompositions of Classifying Spaces with Applications to Algebraic
Cobordism Theories ..................................................... 123
C. HOMOTOPY THEORY; NILPOTENT SPACES; LOCALIZATION
.P Booth, .P Heath and R. Piccinini:
Fibre Preserving Maps and Functional Spaces ............................ 158
.P Booth, .P Heath and .R Piccinini:
Characterizing Universal Fibrations .................................... 168
.P Hilton:
On Orbit Sets for Group Actions and Localization ....................... 185
.P Hilton, .G Mislin, J. Roitberg and .R Steiner:
On Free Maps and Free Homotopies Into Nilpotent Spaces ................. 202
.G Mislin:
Conditions for Finite Domination for Certain Complexes ................. 219
J. Segal:
An Introduction to Shape Theory ........................................ 225
M. Tangora:
Generating Curtis Tables ............................................... 243
VJ
.D GROUP COHOMOLOGY; ACTIONS
A.Liulevicius:
Flag Manifolds and Homotopy Rigidity of Linear Actions .................. 254
D.Sjerve:
Generalized Homological Reduction Theorems .............................. 262
ADDRESSES OF CONTRIBUTORS .................................................... 276
PARTICIPANTS ................................................................. 278
THE UNIQUENESS OF COPRODUCT DECOMPOSITIONS
FOR ALGEBRAS OVER A FIELD f
by
Roy Douglas
.1 Introduction
Coproduct decompositions of various types of graded F-algebras will be
considered, where F is an arbitrary fixed field. The main result will be a
"unique factorlzatlon • " ,T for such decompositions. For some of these types of
algebras, the coproduct is just the appropriate type of tensor product.
We will consider various categories of F-algebras and their associated
commutative semigroups of isomorphism classes of objects (where the binary
semigroup operation is induced by coproduct). The "unique factorization"
results will then be expressed by the statements that the above semigroups are
free commutative semigroups.
Examples of suitable categories of F-algebras are the following:
)i( the category of all connected, finitely generated,
associative, commutative, (graded) F-algebras.
)2( the category of all connected, finitely generated,
associative, graded-commutative, (graded) F-algebras.
)3( the category of all connected, finitely generated,
(graded) Lie algebras over .F
The discussion below proves the "unique factorization" assertion in
examples )i( and (2), )i( being a corollary of (2). Moreover, this discussion
may be generalized to give "unique factorization" results for many other types
of graded algebras, including example (3).
* Research partly supported by the National Research Council of Canada.
f The content of this address is the result of joint work with .R .A Body.
In case F is a perfect field, this result is demonstrated in 2 .
In case F has characteristic zero, certain (large and interesting) classes
of F-homotopy types of topological spaces satisfy a unique decomposition property
with respect to direct product. This is demonstrated for two such classes of
F-homotopy types in 2 and 3 , respectively.
All these results, somewhat reminiscent of the Krull-Schmidt theorem, are
proved by a study of the conjugacy properties of certain linear algebraic groups
of automorphisms.
.2 Unique Factorization for Coproduct~
Let F be an arbitrary field.
~(F) (resp., A(F)) will denote the category whose objects are all
associative, graded-commutative (resp., strictly commutative), connected,
finitely generated F-algebras, and whose morphisms are all degree preserving
F-algebra homomorphisms. For brevity, the objects of ~(F) will be referred
to as F-algebras. The graded-commutative (resp., commutative) tensor product
is the coproduct in ~(F) (resp., A(F)).
The isomorphism classes of objects of Q(F)(resp., A(F))form a
commutative semigroup ~(F)(resp., ~(F)), where the binary operation is
induced by O (the unit is the "zero object" F). The unique factorization of
F-algebras (with respect to ~ ) is expressed by:
Theorem .i m(F) is a free commutative semigroup.
By doubling the gradation degrees we obtain the following
Corollary. ~(F) is a free commutative semigroup.
Before proving Theorem ,i several useful observations will be recorded.
Let A be an F-algebra. A is non-trivial if A ~ F. We say a non-trivial A is
irreducible if A is not the tensor product of two non-trivial F-algebras.
Definition. A finite set {el, ..., e n} of (graded) F-algebra endomorphisms of
A, will be called a splitting (of A) if the following conditions are satisfied:
)i( e. . .e = .e , i = i, ... , n (Idempotent)
1 l l
(2) .e . e. = 0 , i # j (Orthogonal) .
z j
(O is the trivial endomorphism, which factors through the
zero object.)
(3) Each F-algebra, Image (ek) , is irreducible, k = i, ... , n
(4) The canonical msihprom
n
(~ Image(ek) ÷ A
k=l
is an isomorphism in ~(F).
Two splittings of A, {e I .... , en) and {fl' ... , fn ) are said to be
equivalent if, for some permutation u, the F-algebras Image(ei) and
Image(fa(i) ) are isomorphic, i=l, ... , n.
-I -
If ~ is an automorphism of A, then {e I , ... , e n} and {~.el.~ , . . . ~ c~.e .C~
n
are equivalent splittings.
Proposition 2. If {el, ... e } and {fl fm } are splittings of an
F-algebra A and ei.f. = f~.ei,j for all i = i, ..., n and j = i, ... , m,
3
then these splittings are equivalent. ( Thus, n = m . )
(The proof of this proposition is elementary; for details see 2 , Lemma 2.)
Theorem 1 follows easily from Proposition 2 and the following lemma.
Lemma 3. If {el, ... , e n} and {fl ..... fm } are splittings of an F-algebra
A, then there exists an automorphism ~:A ÷ A such
that a.e..a-l.f.= f..~.e..a -I
1 j j 1
for all i = 1, ... , n and j = 1, ... , m.
3. Proof of Lemma 3.
Let End(A) be the semigroup of all endomorphisms of F-algebra A, and let
Aut(A) be the group of invertible elements of End(A).
Since A = ¢ A i is finitely generated as an F-algebra, there is an
i~0
such that A is generated (as an F-algebra) by the finite dimensional F-vector
subspace V = ~ A i
The restriction, End(A) + HomF(V,V), and its restriction Aut(A) ÷ GLF(V )
are injective, representing End(A) and Aut(A) as sets of square matrices with
coefficients in F.
End(A) is the set of zeros of an obvious set of linear and quadratic
polynomials (in the entries of the matrices) with coefficients in F.
Let K be an algebraic closure of F and let W = V ~ K. The above set of
polynomials (with coefficients in F) then defines a variety E in the K-affine
space HomK(W,W ). Moreover, E is a closed set in the Zariski F-topology on
HomK(W,W ). (See 4 for definitions.) Of course, End(A) is the set of
F-rational points of E.
Similarly, Aut(A) is the set of F2rational points in the affine algebraic
group G = E ~ GLK(W ).
There is a finite, purely inseparable extension field L of F F( ~ L #- K),
such that E and G are defined over .L (In case F is perfect, L = F.)
Now consider the splitting
{el, --- , e }n C )A(dnE for F-algebra .A
n
Let A k = Image(ek) , a sub-F-algebra of ,A with ~ A k = .A
K=l=
~ ~ kP @ ln~k kP
A p = A k and V = A k
n n =
E Pk=P k=l Z ~ Z
k=l k=l kp
LTN i(cid:127)l
( Pi
Thus, W = = A i ~ K ) , where the n-fold tensor product
n
Z Pi ~% F
i=l
is constructed over K.
For each i=l,2,...,n and each t e F , the automorphism ~(t)~Aut(A)#" GLF(V )
n
is defined to be scalar multiplication by t s on a direct summand ~ Aj pj
j=l =
of V , where s = Z Pj-
j~i
X I • : K* ÷ G is a one parameter subgroup of G, where Xi is defined similarly in
terms of such an eigenspace decomposition of W.
Let S be the subgroup of G generated by (~i(t) Ite K , i = 1 ..... n}.
Then S is an L-split torus of G. ( cf. 4~, p.200.)
Observe that (el, ... , en}~E, where ~ is the K-Zariski closure of
S in E.
In fact,
e i E Xi(K* ) m E, since K is an infinite field.
Thus, there is a maximal L-split torus Te~G , such that (e I .... ,~n}~Te~E.
Similarly, there is a maximal L-split torus TfCG, such that (fl' "'"fn}~'f GE"
Using an unpublished result of Borel and Tits 5 it follows that there is
an L-rational point ~ E G(L) such that 8 • T e • ~-i = Tf )*(
In case F is a perfect field, we have L = F, and B e G(F) = Aut )A( is the
required automorphism ~ of Lemma .3 This follows from the fact that the closure
of a torus is a commutative set of endomorphisms, and the fact that conjugation
by B is a homeomorphism.
In case F is not perfect, a finite iteration of the Frobenius morphism
takes L into F: ~S(L) ~F.
Let a = ¢s(8) e G(F) = Aut(A) and observe that ~S(ei) = e i c End(A)~ E.
Notice ~S(Te) is dense in Te, since F is infinite. (Recall that finite fields
are perfect.)
Thus, (e I ..... en}~S(Te). Similarly, {fl' ... , fm}~ ~S(Tf). Again,
cS(Te) and ~s(Tf) are each commutative sets of endomorphisms.
Applying ~s to )*( we obtain
• CS(Te) a -I = CS(Tf) which implies the conclusion of
Lemma 3.
Q.E .D.