Algebraic Topology Waterloo 1978: Proceedings of a Conference Sponsored by the Canadian Mathematical Society, NSERC (Canada), and the University of Waterloo, June 1978

667 Pages·1979·9.754 MB·English

by  G. Carlsson, R. James Milgram (auth.), Peter Hoffman, Victor Snaith (eds.)

Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
741
Algebraic Topology
Waterloo 1978
Proceedings of a Conference Sponsored by the
Canadian Mathematical Society, NSERC (Canada),
and the University of Waterloo, June 1978
Edited by
Peter Hoffman and Victor Snaith
Springer-Verlag
Berlin Heidelberg New York 1979
Editors
Peter Hoffman
Department of Pure Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Victor Snaith
Department of Mathematics
University of Western Ontario
London, Ontario
Canada N6A 5B9
AMS Subject Classifications (1980): lOC05, 13Dxx, 16A54,
18 Fxx, 18 G XX, 55-XX, 57-XX, 58Axx
ISBN 3-540-09545-4 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-09545-4 Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data
Main entry under title:
Algebraic topology, Waterloo 1978.
{Lecture notes in mathematics; 741}
Bibliography: p.
Includes index.
1. Algebraic topology--Congresses. I. Hoffman, Peter, 1941-II. Snaith, Victor Percy,
1944-III. Canadian Mathematical Congress {Society} IV. University of Waterloo.
V. Series: Lecture notes in mathematics {Berlin} ; 741.
0A3.L28 no. 741 [QA612) 510'.8s [514'.2)79-20052
ISBN 0-387-09545-4
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© by Springer-Verlag Berlin Heidelberg 1979
2141/3140-543210
FOREWORD
The present volume contains the Proceedings of a conference held
at the University of Waterloo in June of 1978. The conference was
part of a Summer Research Institute of the Canadian Mathematical
Society, and was held simultaneously with a conference in Ring Theory
whose proceedings also appear in this series.
The main emphasis was on L-theory and on topological and alge
braic K-theory. The volume is divided into four sections as indi
cated in the table of contents.
We would like to express our gratitude to all the participants
for their contributions to the conference, and particularly to Ian
Hambleton, Richard Kane, Stan Kochman, Reinhardt Schultz and Rick
Sharpe for help with this volume. We are grateful to Sue Embro for
her work with the manuscripts, and to the sponsoring organizations for
financial assistance.
Waterloo, Canada Peter Hoffman
January 1979 Victor Snaith
CONTENTS
A. L-THEORY AND ALGEBRAIC K-THEORY
G. Carlsson and R.J. Milgram:
The Structure of Odd L-Groups -----------------------------
I. Hambleton and R.J. Milgram:
The Surgery Group L~(Z(G)) for G a Finite 2-group ----- 73
H.J. Munkholm:
Whitehead Torsion for PL Fiber Homotopy Equivalences ------ 90
A. Ranicki:
Localization in Quadratic L-theory ------------------------ 102
R.W. Sharpe:
K (Z[Z/S]) is Generated by Relations Among 2x2 Matrices - 158
2
L. Taylor and B. Williams:
Surgery Spaces: Formulae and Structure -------------------- 170
B. GROUP ACTIONS
D.R. Anderson and I. Hambleton:
Balanced Splittings of Semi-free Actions on Homotopy
Spheres --------------------------------------------------- 196
A.H. Assadi:
Some Examples of Finite Group Actions --------------------- 206
T. torn Dieck and T. Petrie:
The Homotopy Structure of Finite Group Actions on Spheres - 222
K.H. Dovermann:
Addition of Equivariant Surgery Obstructions -------------- 244
J. Ewing:
Obstructions to Realizing Equivariant Witt Classes -------- 272
R. Lashof:
Stable G-Smoothing ---------------------------------------- 283
A. Liulevicius:
Linear Actions on Friendly Spaces ------------------------- 307
VI
J. McLeod:
The Kunneth Formula in Equivariant K-theory --------------- 316
R. Schultz:
Isotopy Classes of Periodic Diffeomorphisms on Spheres ---- 334
C. HOMOTOPY AND COHOMOLOGY
J.M. Boardman:
Original Brown-Peterson Spectra --------------------------- 355
D.M. Davis:
BP-Operations and Mappings of Stunted Complex Projective
Spaces 373
S.O. Kochman and V.P. Snaith:
On the Stable Homotopy of Symplectic Classifying and
Thorn Spaces 394
P.S. Landweber:
New Applications of Commutative Algebra to Brown-Peterson
Homology -------------------------------------------------- 449
P.S. Landweber:
The Signature of Symplectic and Self-Conjugate Manifolds -- 461
D. Sjerve:
Homology Isomorphisms -------------________________________ 473
R. Steiner:
An Isomorphism Between Products of Abelian Groups --------- 512
D. LOOP AND H-SPACES
D.W. Anderson:
Axiomatic Homotopy Theory --------------------------------- 520
J.R. Harper:
Construction of mod pH-spaces II ------------------------- 548
R. Kane:
The Mod 3 Cohomology of the Exceptional Lie Group ES ------ 557
D. Kraines and T. Lada:
A Counterexample to the Transfer Conjecture --------------- 588
J.P. May:
Infinite Loop Space Theory Revisited ---------------------- 625
R. Seymour and V. Snaith:
A J-homomorphism Associated with a Space of Empty
Varieties (addenda and corrigenda to two papers on
the J-homomorphism) --------------------------------------- 643
ADDRESSES OF CONTRIBUTORS --------------------------------------- 653
LIST OF PARTICIPANTS -------------------------------------------- 655
WATERLOO TOPOLOGY SRI
List of Talks
May 15 R. Kane
16 V. Snaith On Algebraic Vector Bundles over
Number Fields.
17 R. Sharpe Introduction to L-theory
R. Steiner Infinite loop constructions I
18 K. Murasugi On Representations of Knot Groups
19 R. Steiner Infinite loop constructions II
May 23 I. Hambleton Hermitian Forms and Manifolds
24 Doug Anderson Immersion Theory
25 V. Snaith Introduction to the + construction
and higher K-theory
31 Doug Anderson The Immersion Approach to Triangulations
June 1 I. Hambleton The + construction and homology spheres
2 T. Petrie Background for Smooth Lie Group Actions
5 T. Petrie G surgery and semiclassical applications I
R. Kane BP theory and Finite H-spaces
J. Allard Sums of Stably Trivial Vector Bundles
6 w. Pardon Examples and Applications of Surgery
Obstructions
R. Lashof Stable G-Smoothing
H. Munkholm Finiteness Obstructions, Whitehead
Torsion and Transfer in Algebraic
K-theory
R. Kulkarni Proper Actions of non-compact Groups
and Relativistic Space Forms
B. Williams Formal Surgery Theory
7 P. May Segal's maps, Kahn's maps and Mahowald's
Theorem
I. Madsen Spherical Space Forms
T. Petrie G surgery and semiclassical applications II
H. Dovermann Structure of the Set (Grouplof Equivariant
Surgery Obstructions and Applications
-- J. Harper Construction of mod pH-spaces
x
List of Talks (con'd)
June 8 M. Rothenberg Equivariant Homotopy Type of Spheres
R. Lee Unstable cohomology of SL (Z)
n n
W. Hsiang 1Ti Diff (M )
I. Hambleton Decompositions of Semi-Free Actions
on Homotopy Spheres
J. Ewing Realizing Witt classes
9 P. Zvengrowski Skewness of r-fields on Spheres
and Projective Stiefel Manifolds
J. Milgram The Swan Finiteness Obstruction
T. Lada A Counterexample to the Transfer
Conjecture
K. Murasugi PSL (2,p) Coverings of Knot Manifolds
S. Thomeier Join Constructions and Whitehead
Products
-- G. Carlsson Equivariant Embeddings in Homotopy
Spheres
June 12 D. Kraines Applications of the Delooping Spectral
Sequence
I. Madsen Tangential Homotopy Equivalences
W. Hsiang The Topological Euclidean Space Form
Problem
R. Schultz Compact fibrings of minimal homogeneous
Spaces
D. Gottlieb Lefschetz Numbers of Equivariant Maps
13 A. Ranicki Equivariant Wu classes
J. Milgram Odd Wall Groups
Doug Anderson Triangulations of Locally Triangulable
Spaces
R. Sharpe Strong Euclidean Rings and K2 (Z[Z/5])
S. Lomonaco Homotopy Groups of Knots
14 A. Bak Surgery Groups; a survey of computations
with emphasis on ker(L2~(1T)'" L2~(1T»)
XI
List of Talks (con'd)
June 15 A. Liulevicius Linear Actions on Friendly Spaces
A. Rani cki Poincar~ complexes & Surgery
M. Boardman Ye Olde Original Brown-Peterson
Spectra
R. Patterson Square Preserving Algebra Endomorphisms
of H*(BO;Z/2)
--
K. Varadarajan Finiteness Obstruction for Nilpotent
Spaces
16 K. Ehrlich A Transfer Map in Algebraic K-theory
v. Snaith C*-extensions of Compact Lie Groups
D. Sjerve Homology Isomorphisms and the failure
of Whitehead's Theorem
-- D. Davis BP operations and maps of stunted
complex projective spaces
June 19 P. Landweber New Applications of Commutative Algebra
to MU and BP
20 v. Snaith On Kj.Z/4)
P. Landweber The Signature of Symplectic and Self
Conjugate Manifolds
21 Don Anderson Homotopy Coherence of Functors
S. Kochman The Symplectic Cobordism Ring I
22 S. Kochman The Symplectic Cobordism Ring II
THE STRUCTURE OF ODD L-GROUPS
by
G. Carlsson
R. James Milgram*
In this paper we apply the exact sequence of [C-Mj or [Pj to obtain
extensive information about odd surgery groups. In particular, we completely
determine the proper surgery groups L~(rr,l) [~j for rr a finite 2-group.
Using these groups we then obtain a system of generators for the Wall surgery
h
groups L3 (rr, 1) for rr as above from the exact sequence
For rr a 2-group the 2 primary part of Ko(Z(rr» is determined as the
quotient of a finite group depending only on the rational representation
ring of rr, ~(rr). Then im(j) is easily calculated, and Ker(a) is
the only datum which we presently lack in studying A.
Our main result on finite 2-groups (Theorem 4.9), is
Theorem B: Let n(rr) be the number of summands M (F) contained in
n
i
q) (rr) where F is a real field, then
L~(rr,l) (Z/2)n(rr)-1
2
For example, let rr be the dihedral group Z/ i x Z/2(t = tgtg
T
2
*Research supported in part by NSF MCS77-01623 and NSF MCS76-0146-A01