Lecture Notes in Mathematics

Lecture Notes in Mathematics 1217 Transformation Groups Poznar~ 1985 Proceedings of a Symposium held in Poznar~, July 5-9, 1985 Edited by S. Jackowski and K. Pawa owski IIIII HI I I III Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editors Stefan Jackowski Pafac Kultury i Nauki IXp. 00-901 Warszawa, Poland Krzysztof Pawalowski Uniwersytet ira. A. Mickiewicza w Poznaniu 60-769 PoznaS, Poland Mathematics Subject Classification (1980): 57 S XX; 57 S 10; 57 S 15; 57 S 17; 57S25; 57R67; 57R80; 20J05 ISBN 3-540-16824-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16824-9 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr, 2146/3140-543210

Dedicated to the memory of A Jankowski and W. Pulikowski

P R E F A C E The Symposium on Transformation Groups supported by the Adam Mickiewicz University in Pozna~ was held in Pozna~, July 5-9, 1985. The symposium was dedicated to the memory of two of our teachers and friends, Andrzej Jankowski and Wojtek Pulikowski on the tenth anniversary of their deaths. These proceedings contain papers presented at the symposium and also papers by mathematicians who were invited to the meeting but were unable to attend. All papers have been refereed and are in their final forms. We would like to express our gratitude to the authors and the many referees. The participants and in particular the lecturers contributed to the success of the symposium and we are most grateful to all of them. Special thanks are due to our colleagues Ewa Marchow, Wojtek Gajda, Andrzej Gaszak, and Adam Neugebauer for their help with the organizational work and to Barbara Wilczy~ska who handled the administrative and secretarial duties. The second editor thanks Sonderforschungsbereich 170 in G6ttingen for its hospitality which was very helpful in the preparation of the present volume. Finally, we would like to thank Marrie Powell and Christiane Gieseking for their excellent typing. Pozna~/Warszawa, 20.O6.1986

ANDRZEJ JANKOWSKI (1938-1975) WOJCIECH PULIKOWSKI (1947-1975) Andrzej graduated in 1960 from the Nicolaus Copernicus University in Torud. Topology was his passion and his interests were very broad. Andrzej worked on algebraic and differential topology, his main papers being concerned with operations in generalized cohomology theories and with formal groups. His was not an easy task. Andrzej worked essentially alone. Polish topologists were at that time continuing the tradition of their pre-war school. Andrzej's friend and Ph.D. student wrote*): "He wanted to understand the deepest and most difficult theorems found by his contemporaries. At that beautiful time of great discoveries Andrzej faced the difficult obstacle of being alone. He put a lot of effort into overcoming this difficulty, and also conveying his knowledge to others." Andrzej began to lecture on algebraic topology and to organize seminars as soon as he joined the University of Warsaw in 1962. For nine years, from 1967, he was the spiritusmovens of the Summer School on Algebraic Topology held annually in Gda~sk. He moved to Gda~sk in 1971. From 1969 until his death he led a seminar on transformation groups. Wojtek Pulikowski was one of the participants. Wojtek graduated in 1969 from Pozna~ and moved to Warszawa and then to Gda~sk. In 1973 Wojtek obtained his Ph.D. for the work on equivariant bordism theories indexed by representations and returned to Pozna~. He invested great effort into organizing seminars, summer schools and meetings on various topics in algebraic and differential topology. At the same time he continued teaching his students and before long directed their research towards transformation groups. Wojtek was a born teacher, able to convey not only his knowledge but also his passion, enthusiasm, and interest in the subject. He wrote a number of papers on equivariant homology theories, but he spent most of his time in teaching - which he did with joy and love. His friends and students all owe him a great deal. Besides mathematics, both Andrzej and Wojtek had another passion - mountains. And in the mountains both of them met their death in August 1975, Andrzej climbing the Tirach Mir peak in the Hindu Kush mountains and Wojtek in an accident in the Beskidy mountains in Poland. *) R.Rubinsztein: "Andrzej Jankowski (1938-1975)", Wiadomo~ci Mate- matyczne, vol. XXIII (1980), pp. 85-91.

TABLE OF CONTENTS Chronological list of talks... Current addresses of authors and participants... XII XI Allday,C., and Puppe,V.: Bounds on the torus rank... 1 Andrzejewski,P.: The equivariant Wall finiteness obstruction and Whitehead torsion... ii Assadi,A.: Homotopy actions and cohomology of finite groups... 26 Assadi,A.: Normally linear Poincar6 complexes and equivariant splittings... 58 Carlsson,G.: Free (Z/2)k-actions and a problem in commutative algebra... 79 tom Dieck,T. und L6ffler,P.: Verschlingungszahlen von Fixpunktmengen in Darstellungsformen. II... 84 Dovermann,K.H., and Rothenberg,M.: An algebraic approach to the generalized Whitehead group... 92 Battori,A.: Almost complex Sl-actions on cohomology complex projective spaces... 115 Illman,S.: A product formula for equivariant Whitehead torsion and geometric applications... 123 Jaworowski,J.: Balanced orbits for fibre preserving maps of S 1 and S 3 actions... 143 Kania-Bartoszy~ska,J.: Involutions on 2-handlebodies... 151 Katz,G.: Normal combinatorics of G-actions on manifolds... 167 Kawakubo,K.: Topological invariance of equivariant rational Pontrjagin classes... 183 Ko{niewski,T.: On the existence of acyclic F complexes of the lowest possible dimension... 196 Laitinen,E.: Unstable homotopy theory of homotopy representations... 210 Liulevicius,A., and Ozaydin,M.: Duality in orbit spaces... 249 Marciniak,Z.: Cyclic homology and idempotents in group rings... 253 Masuda,M.: ~2 surgery theory and smooth involutions on homotopy complex projective spaces... 258 Matumoto,T., and Shiota,M.: Proper subanalytic transformation groups and unique triangulation of the orbit spaces... 290

May,J.P.: A remark on duality and the Segal conjecture... 303 Pedersen,E.K.: On the bounded and thin h-cobordism theorem parametrized by ~k... 306 Ranicki,A.: Algebraic and geometric splittings of the K- and L-groups of polynomial extensions... 321 Schw~nzl,R., and Vogt,R.: Coherence in homotopy group actions 364 Szczepa6ski,A.: Existence of compact flat Riemannian manifolds with the first Betti number equal to zero... 391 Weintraub,S.H.: Which groups have strange torsion?... 394

CHRONOLOGICAL LIST OF TALKS A.Liulevicius (Chicago): Duality of symmetric powers of cycles S.Illman (He]sinki): P.L6ffler (G6ttingen): W.Marzantowicz A.Szczepa~ski (Gda~sk): (Gda6sk): W.Browder (Princeton): V.Puppe (Konstanz): Z.Marciniak (Warszawa): E.Laitinen (Helsinki): J.Kania-Bartoszy~ska (Warszawa): R.Vogt (Osnabr~ck): K.H.Dovermann (West Lafayette): M.Lewkowicz A.Assadi (Wroclaw): (Charlottesville): E.K.Pedersen (Odense): M.Sadowski J.Tornehave (Gda~sk): (Aarhus): S.Weintraub (Baton Rouge): K.Pawa~owski (Pozna~): R.Oliver (Aarhus): S.Jackowski (Warszawa): J.Ewing (Bloomington): Product formula for equivariant Whitehead torsion Realization of exotic linking numbers of fixed point sets in representation forms The Sl-equivariant topology and periodic solutions of ordinary differential equations Euclidean space forms with the first Betti number equal to zero Actions on projective varieties Bounds on the torus rank Idempotents in group rings and cyclic homology Unstable homotopy theory of homotopy representations Classification of involutions on 2-handlebodies Coherence theory and group actions Symmetries of complex projective spaces Nonabelian Lie group actions and positive scalar curvature Homotopy actions and G-modules The bounded and thin h-cobordism theorems Injective sl-actions on manifolds covered by ~n Units in Burnside rings and the Kummer theory pairing Group actions and certain algebraic varieties Smooth group actions on disks and Euclidean spaces A transfer map for compact Lie group actions A fixed point theorem for p-group actions Symmetries of surfaces and homology

CURRENT ADDRESSES OF AUTHORS AND PARTICIPANTS Christopher Allday University of Hawaii at Manoa Honolulu, HI 96822, USA Pawe[ Andrzejewski Uniwersytet Szczeci~ski ul. Wielkopolska 15 70-451 Szczecin, Poland Amir H. Assadi University of Wisconsin Madison, WI 53706, USA Grzegorz Banaszak Uniwersytet Szczeci6ski ul. Wielkopolska 15 70-451 Szczecin, Poland Agnieszka Bojanowska OO-901 Warszawa, Poland William Browder Princeton University Princeton, NJ 08544, USA Gunnar Carlsson Princeton University Princeton, NJ 08544, USA Tammo tom Dieck Mathematisches Institut Universit~t G6ttingen BunsenstraSe 3-5 3400 G6ttingen, West Germany Ryszard Doman Uniwersytet im. A.Mieckiewicza 60-769 Pozna, Poland Karl Heinz Dovermann University of Hawaii at Manoa Honolulu, HI 96822, USA John Ewing Indiana University Bloomington, IN 47405, USA Wojciech Gajda 60-769 Pozna~, Poland Andrzej Gaszak 60-769 Pozna, Poland Jean-Pierre Haeberly University of Washington Seattle, WA 98195, USA Akio Hattori Faculty of Science University of Tokyo Hongo, Tokyo, 113 Japan S6ren Illman University of Helsinki Hallituskatu 15 OOIOO Helsinki io, Finland Stefan Jackowski OO-901 Warszawa, Poland Tadeusz Januszkiewicz Uniwersytet Wroclawski Pl. Grunwaldzki 2/4 50-384 Wroc!aw, Poland

XIIl Jan Jaworowski Indiana University Bloomington, IN 47405, USA Joanna Kania-Bartoszylska University of California Berkeley, CA 94720, USA Gabriel Katz Ben Gurion University Beer-Sheva 84105, Israel Katsuo Kawakubo Osaka University Toyonaka, Osaka, 560 Japan Tadeusz Ko~niewski OO-901 Warszawa, Poland Piotr Kraso6 Uniwersytet Szczecilski ul. Wielkopolska 15 70-451 Szczecin, Poland Erkki Laitinen University of Helsinki Hallituskatu 15 OOIOO Helsinki I0, Finland Marek Lewkowicz Uniwersytet Wroclawski PI. Grunwaldzki 2/4 50-384 Wroclaw, Poland Arunas Liulevicius University of Chicago Chicago, IL 60637, USA Peter L6ffler Mathematisches Institut Universit~t G6ttingen Bunsenstr. 3-5 3400 G6ttingen, West Germany Ewa Marchow 60-769 Pozna~, Poland Zbigniew Marciniak 00-901 Warszawa, Poland Waclaw Marzantowicz Uniwersytet Gdalski ul. Wita Stwosza 57 80-952 Gda~sk, Poland Mikiya Masuda Osaka City University Osaka 558, Japan Takao Matumoto Department of Matheamtics Faculty of Science Hiroshima University Hiroshima 730, Japan J. Peter May University of Chicago Chicago, IL 60637, USA Janusz Migda 60-769 Poznal, Poland Adam Neugebauer 60-769 Pozna6, Poland Krzysztof Nowilski OO-901 Warszawa, Poland Robert Oliver Matematisk Institut Aarhus Universitet Ny Munkegade 8000 Aarhus C, Denmark

XIV Murad Ozaydin University of Wisconsin Madison, WI 53706, USA Krzysztof Pawalowski 60-769 Pozna~, Poland Erik Kjaer Pedersen Matematisk Institut Odense Universitet Campusvej 55 5230 Odense M, Denmark Jerzy Popko Uniwersytet Gda6ski ul. Wita Stwosza 57 80-952 Gda~sk, Poland J6zef Przytycki OO-901 Warszawa, Poland Volker Puppe Fakult~t fur Mathematik Universit~t Konstanz Postfach 5560 7750 Konstanz, West Germany Andrew Ranicki Edinburgh University King's Buildings, Mayfield Rd. Edinburgh EH9 3JZ, Scotland, UK Michal Sadowski Uniwersytet Gda~ski ul. Wita Stwosza 57 80-952 Gda~sk, Poland Jan Samsonowicz Politechnika Warszawska PI. Jedno ci Robotniczej 1 00-661 Warszawa, Poland Roland SchwAnzl Fachbereich Mathematik Universit~t Osnabr~ck AlbrechtstraSe 28 4500 OsnabrQck, West Germany Masahiro Shiota Faculty of General Education Nagoya University Nagoya 464, Japan Jolanta Sgomi~ska Uniwersytet im. M.Kopernika ul. Chopina 12 87-1OO Toru~, Poland Andrzej Szczepa~ski Politechnika Gda~ska ul. Majakowskiego 11/12 80--952 Gda~sk, Poland J~rgen Tornehave Matematisk Institut Aarhus Universitet Ny Munkegade 8000 Aarhus C, Denmark Martin Raussen Pawel Traczyk Institut for Elektroniske Systemer Aalborg Universitetscenter Strandvejen 19 9000 Aalborg, Denmark OO-901 Warszawa, Poland Melvin Rothenberg University of Chicago Chicago, IL 60637, USA Slawomir Rybicki Politechnika Gda ska ul. Majakowskiego 11/12 80-952 Gda6sk, Poland Rainer Vogt Fachbereich Mathematik Universitit Osnabr~ck AlbrechtstraSe 28 4500 Osnabr~ck, West Germany Steven H. Weintraub Louisiana State University Baton Rouge, LA 70803, USA