SURGERY AND GEOMETRIC TOPOLOGY Andrew Ranicki and Masayuki Yamasaki Editors Proceedings of a conference held at Josai University Sakado September Contents Foreword v Program of the conference vii List of participants ix Toshiyuki Akita Cohomology and Euler characteristics of Coxeter groups Frank Connolly and Bogdan Vajiac Completions of StratiedEnds Zbigniew Fiedorowicz and Yong jin Song The braid structure of mapping class groups Bruce Hughes Control led topological equivalence of maps in the theory of stratied spaces and approximate brations Tsuyoshi Kato The asymptotic method in the Novikov conjecture Tomohiro Kawakami Anoteonexponential ly Nash G manifolds and vector bund les Masaharu Morimoto On xedpoint data of smooth actions on spheres Eiji Ogasa Some results on knots and links in al l dimensions Erik K Pedersen Control led algebra and topology Frank Quinn Problems in lowdimensional topology Andrew Ranicki slides on chain duality Andrew Ranicki and Masayuki Yamasaki Control led Ltheory preliminary announce ment iii iv CONTENTS John Ro e Notes on surgery and C algebras Tatsuhiko Yagasaki Characterizations of innitedimensional manifold tuples and their applications to homeomorphism groups Foreword As the second of a conference series in mathematics Josai University sp onsored the Conference on Surgery and Geometric Top ology during the week of September The scientic program consisted of lectures listed b elow and there was an excursion to Takaragawa Gunma on the st This volume collects pap ers by participantsaswell as some of the abstracts pre pared by the lecturers for the conference The articles are also available electronically on WWW from httpmathjosaiacjpyamasakiconferencehtml at least for several years Wewould like to thank Josai University and GrantinAid for Scientic Research A of the Ministry of Education Science Sp orts and Culture of Japan for their generous nancial supp ort Thanks are also due to the university sta for their various supp ort to the lecturers and to the participants of the conference Andrew Ranicki Edinburgh Scotland Masayuki Yamasaki Sakado Japan Decemb er v Program of the Conference Surgery and Geometric Top ology Sakado Saitama Septemb er Tuesday Andrew Ranicki Chain Duality Erik Pedersen Simplicial and Continuous Control Toshiyuki Akita Cohomology and Euler Characteristics of Coxeter Groups Septemb er Wednesday Frank Quinn Problems with Surgery and Handleb o dies in Low Dimensions John Ro e Surgery and Op erator Algebras Tsuyoshi Kato The Asymptotic Metho d in the Novikov Conjecture Eiji Ogasa On the Intersection of Spheres in a Sphere Septemb er Thursday Bruce Hughes Stratied Spaces and Approximate Fibrations Francis Connolly An End Theorem for Manifold Stratied Spaces TatsuhikoYagasaki Innitedimensional Manifold Triples of Homeomorphism Groups vii viii PROGRAM OF THE CONFERENCE Septemb er Friday Yong jin Song The Braid Structure of Mapping Class Groups Masaharu Morimoto On Fixed Point Data of Smo oth Actions on Spheres Masayuki Yamasaki Epsilon Control and Perl List of Participants The numb ers corresp ond to the keys to the photograph on page xi Toshiyuki Akita Fukuoka Univ akitassatfukuokauacjp Yuko Asada Nagoya Univ Naoaki Chinen Univ of Tsukuba Frank Connolly Univ of Notre Dame USA connollyndedu Ko ji Fujiwara Keio Univ fujiwaramathkeioacjp Hiro o Fukaishi Kagawa Univ fukaishiedkagawauacjp Hiroaki Hamanaka Kyoto Univ Tetsuya Hosaka Univ of Tsukuba Bruce Hughes Vanderbilt Univ USA hughesmathvanderbiltedu Kazumasa Ikeda TokyoUniv nikedahongoeccutokyoacjp Jun Imai NTT Communication Sci Lab imaicslabkeclnttjp p Hiroyuki Ishibashi Josai Univ hishimathjosaiacj Hiroyasu Ishimoto Kanazawa Univ ishimotokappaskanazawauacjp Osamu Kakimizu Niigata Univ kakimizuedniigatauacjp Tsuyoshi Kato Kyoto Univ tkatokusmkyotouacjp Tomohiro Kawakami Osaka Pref College lsipcosakapctacjp Kazuhiro Kawamura Univ of Tsukuba kawamuramathtsukubaacjp Takahiro Kayashima Kyushu Univ kayapmathkyushuuacjp Yukio Kuribayashi Tottori Univ Yasuhiko Kitada Yokohama National Univ kitadamathlabsciynuacjp Minoru Kobayashi Josai Univ mkobamathjosaiacjp Yukihiro Ko dama Kanagawa Univ Akira Koyama OsakaKyoiku Univ koyamaccosakakyoikuacjp Kenichi Maruyama Chiba University maruyamacueechibauacjp Mikiya Masuda OsakaCityUniv masudascihimejitechacjp Mamoru Mimura Okayama Univ mimuramathokayamauacjp Kiyoshi Mizohata Josai University mizohatamathjosaiacjp Masaharu Morimoto Okayama Univ morimotomathemsokayamauacjp Yutaka Morita Josai Univ ymorieuclidesjosaiacjp ix x LIST OF PARTICIPANTS Hitoshi Moriyoshi Hokkaido Univ moriyosimathhokudaiacjp Mitsutaka Murayama TokyoInstTech murayamamathtitechacjp Ikumitsu Nagasaki Osaka Univ nagasakimathsciosakauacjp Masatsugu Nagata Kyoto Univ RIMS nagatakurimskyotouacjp Kiyoko Nishizawa Josai Univ kiyokomathjosaiacjp Eiji Ogasa Tokyo Univ imunixccutokyoacjp Jun OHara Tokyo Metrop olitan Univ oharamathmetrouacjp Hiroshi Ohta Nagoya Univ ohtamathnagoyauacjp Nobuatsu Oka Aichi Kyoiku Univ Kaoru Ono Oc hanomizu Univ onomathochaacjp Erik Pedersen SUNY at Binghamton USA erikmathbinghamtonedu Frank Quinn Virginia Tech USA quinnmathvtedu Andrew Ranicki Edinburgh Univ UK aarmathsedacuk John Ro e Oxford Univ UK jroejesusoxacuk Katsusuke Sekiguchi Kokushikan Univ Kazuhisa Shimakawa Okayama Univ kazumathokayamauacjp Yong jin Song Inha Univ Korea yjsongmathinhaackr Toshio Sumi Kyushu Institute of Design sumikyushuidacjp Toshiki Suzuki Josai Univ Hideo Takahashi NagaokaInstTech Junya Takahashi Tokyo Univ Tatsuru Takakura Fukuoka Univ takakurassatfukuok auacjp Dai Tamaki Shinshu Univ rivulusgipacshinshuuacjp Fumihiro Ushitaki Kyoto Sangyo Univ ushitakiksuvxkyotosuacjp TatsuhikoYagasaki Kyoto Inst Tech yagasakiipckitacjp Yuichi Yamada Tokyo Univ yyyamadamsutokyoacjp Hiroshi Yamaguchi Josai Univ hyamamathjosaiacjp TsuneyoYamanoshita Musashi Inst Tech Masayuki Yamasaki Josai Univ yamasakimathjosaiacjp Katsuya Yokoi Univ of Tsukuba yokoimathtsukubaacjp Zenichi Yoshimura NagoyaInstTech yosimuramathkyy nitechacjp Surgery and Geometric Top ology Contents Foreword v Program of the conference vii List of participants ix Toshiyuki Akita Cohomology and Euler characteristics of Coxeter groups Frank Connolly and Bogdan Vajiac Completions of StratiedEnds Zbigniew Fiedorowicz and Yong jin Song The braid structure of mapping class groups Bruce Hughes Control led topological equivalence of maps in the theory of stratied spaces and approximate brations Tsuyoshi Kato The asymptotic method in the Novikov conjecture Tomohiro Kawakami Anoteonexponential ly Nash G manifolds and vector bund les Masaharu Morimoto On xedpoint data of smooth actions on spheres Eiji Ogasa Some results on knots and links in al l dimensions Erik K Pedersen Control led algebra and topology Frank Quinn Problems in lowdimensional topology Andrew Ranicki slides on chain duality Andrew Ranicki and Masayuki Yamasaki Control led Ltheory preliminary announce ment John Ro e Notes on surgery and C algebras Tatsuhiko Yagasaki Characterizations of innitedimensional manifold tuples and their applications to homeomorphism groups COHOMOLOGY AND EULER CHARACTERISTICS OF COXETER GROUPS TOSHIYUKI AKITA Introduction Coxeter groups are familiar ob jects in many branches of mathematics The connections with semisimple Lie theory have b een a ma jor motivation for the study of Coxeter groups Crystallographic Coxeter groups are involved in KacMo o dy Lie algebras which generalize the entire theory of semisimple Lie algebras Coxeter groups of nite order are known to be nite reection groups which app ear in invariant theory Coxeter groups also arise as the transformation groups generated by reections on manifolds in a suitable sense FinallyCoxeter groups are classical ob jects in combinatorial group theory In this pap er we discuss the cohomology and the Euler characteristics of nitely generated Coxeter groups Our emphasis is on the role of the parab olic subgroups of nite order in b oth the Euler characteristics and the cohomology of Coxeter groups The Euler characteristic is dened for groups satisfying a suitable cohomological niteness condition The denition is motivated by top ology but it has applications to group theory as well The study of Euler characteristics of Coxeter groups was initiated by JP Serre who obtained the formulae for the Euler characteristics of Coxeter groups as well as the relation b etween the Euler characteristics and the Poincare series of Coxeter groups The formulae for the Euler characteristics of Coxeter groups were simplied byIM Chiswell From his result one knows that the Euler characteristics of Coxeter groups can b e computed in terms of the orders of parab olic subgroups of nite order On the other hand for a Coxeter group W the family of parab olic subgroups of nite order forms a nite simplicial complex F W In general given a simplicial complex K the Euler characteristics of Coxeter groups W with F W K are b ounded but are not unique However it follows from the result of M W Davis nsphere Theorem Inspired that eW if F W is a generalized homology by this result the author investigated the relation b etween the Euler characteris tics of Coxeter groups W and the simplicial complexes F W and obtained the following results If F W is a PLtriangulation of some closed nmanifold M then M eW If F W is a connected graph then eW F W where denotes the genus of the graph TOSHIYUKI AKITA See Theorem