Selected Title s i n Thi s Subserie s

50 V . Turae v an d A . Vershik , Editors , Topology , ergodi c theory , rea l algebrai c geometry .

Rokhlin's memoria l (TRANS2/202 )

49 Michae l Semenov-Tian-Shansky , Editor , L . D . Faddeev' s semina r o n mathematica l

physics (TRANS2/201 )

48 L . Lerman , G . Polotovskii , an d L . Shilnikov , Editors , Method s o f qualitativ e theor y

of differentia l equation s an d relate d topic s (TRANS2/200 )

47 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s way .

Prom probabilit y theor y t o statistica l physic s (TRANS2/198 )

46 Vladimi r Arnold , M a x i m Kontsevich , an d Anto n Zorich , Editors , Pseudoperiodi c

topology (TRANS2/197 )

45 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinste in , Editors , Norther n

California symplecti c geometr y semina r (TRANS2/196 )

44 Alexande r Astashkevic h an d Serg e Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y Collection) (TRANS2/194 )

43 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematica l

Physics (TRANS2/191 )

42 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

(TRANS2/190)

41 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection ) (TRANS2/189 )

40 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

(TRANS2/187)

39 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations (TRANS2/186 )

38 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) (TRANS2/185 )

37 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s (Dedicated t o Seli m Krein ) (TRANS2/184 )

36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 )

35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )

34 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n

the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

(TRANS2/177)

31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Net sve taev , Editors , Mathematic s

in St . Petersbur g (TRANS2/174 )

29 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pes in , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 ) 26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B . Dynkin's Semina r (TRANS2/169 ) 25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )

i«i^Mw

Vladimir Abramovic h ROKHLIN 1919-1984

Topology, Ergodic Theory , Real Algebraic Geometr y Rokhlin's Memoria l

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 20 2

Advances in the Mathematical Sciences—50 (Formerly Advances in Soviet Mathematics)

Topology, Ergodic Theory , Real Algebraic Geometr y

Rokhlin's Memoria l

V. Turae v A. Vershi k Editors

American Mathematica l Societ y '•? Providence , Rhod e Islan d

Ut5?DED

http://dx.doi.org/10.1090/trans2/202

ADVANCES I N TH E MATHEMATICA L SCIENCE S EDITORIAL COMMITTE E

V. I. ARNOLD S. G. GINDIKIN V. P. MASLOV

ABSTRACT. Vladimi r Abramovic h Rokhli n (Augus t 23 , 1919-Decembe r 4 , 1984 ) wa s a n outstand -ing Russia n mathematicia n whos e contribution s t o topolog y an d ergodi c theor y hav e bee n inter -nationally recognized . Th e presen t volum e contain s paper s b y Rokhlin' s colleague s an d students ; the topic s covere d b y thes e paper s reflec t th e wid e spectru m o f Rokhlin' s interest s (topology , geometry, rea l algebrai c geometry , dynamica l systems) .

The boo k i s useful t o researchers an d graduat e student s working i n topology, algebrai c geometry , and dynamica l systems .

2000 Mathematics Subject Classification. P r i m a r y 14P25 , 37Axx ; Secondary 28Axx , 28Dxx , 57Q25 .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-2740- 5

ISSN 0065-929 0

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Contents

Foreword

Vladimir Abramovic h Rokhli n A. M . VERSHI K

V. A . Rokhli n an d th e moder n theor y o f measurable partition s A. M . VERSHI K

Higher skei n modules , I I J0RGEN ELLEGAAR D ANDERSE N AN D VLADIMI R TURAE V

Betti number s o f rea l numerica l quinti c surface s F. BlHA N

Markov averagin g an d ergodi c theorem s fo r severa l operator s ALEXANDER BUFETO V

Ergodicity o f p-adi c multiplication s an d th e distributio n o f Fibonacc i numbers ZAQUEU COELH O AN D WILLIA M PARR Y

On th e Pontryagin-Vir o for m ALEXANDER DEGTYARE V

Counting zero s o f closed 1-form s MICHAEL FARBE R

Rokhlin's questio n an d smoot h quotient s b y comple x conjugatio n o f singula r real algebrai c surface s SERGEY FINASHI N

On th e numbe r o f even oval s o f a nonsingula r curv e o f even degre e i n R P 2

ILIA ITENBER G

Isometries o f Teichmiille r space s fro m th e poin t o f view o f Mosto w rigidit y NIKOLAI V . IVANO V

Equivalence relation s wit h amenabl e leave s nee d no t b e amenabl e VADIM A . KAIMANOVIC H

Maximal smoothing s o f rea l plan e curv e singula r point s V . KHARLAMOV , J . - J . RlSLER , AN D E . SHUSTI N

viii C O N T E N T S

On th e topologica l pressur e fo r rando m bundl e transformation s YURI KIFE R

Complex orientation s o f M-curve s o f degree 7 S. Yu . OREVKO V

Poincare serie s an d compariso n theorem s fo r variabl e negativ e curvatur e MARK POLLICOT T AN D RICHAR D SHAR P

Encomplexing th e writh e OLEG VIR O

Monotileable amenabl e group s BENJAMIN WEIS S

Spin bordis m o f oriented manifold s an d th e Hauptvermutun g fo r 6-manifold s A. V . ZHUB R

Foreword

Vladimir Abramovic h Rokhli n (Augus t 23 , 1919-Decembe r 4 , 1984 ) wa s a n outstanding Russia n mathematician . H e considerabl y influence d th e developmen t of mathematic s i n Russi a i n th e perio d fro m 195 0 t o th e earl y '80's , especiall y i n Moscow an d Leningra d (St . Petersburg) . Rokhlin' s contribution s t o topolog y an d ergodic theor y hav e bee n internationall y recognized .

Rokhlin studie d mathematic s a t Mosco w Universit y i n th e lat e 1930's . Amon g his professors wer e P. S. Aleksandrov, I . M. Geffand, A . N. Kolmogorov, A . N. Ples-ner, an d L . S . Pontryagin .

Rokhlin successfull y worke d i n severa l area s o f mathematics . Hi s paper s o n three-dimensional an d four-dimensiona l topology , cobordis m theory , rea l algebrai c geometry, an d ergodic theory have become classics. Fo r a discussion of his contribu -tions t o topolog y an d dynamica l systems , se e Uspekh i Mat . Nau k (Russia n Math . Surveys (2000) , no . 4) ; Ergodi c Theor y an d Dynamica l System s 9 (1989) , no . 4 ; and th e boo k A la Recherche de la Topologie Perdue, b y L . Guillo u an d A . Mari n (Birkhauser, 1986) . Se e also the paper s i n thi s volume .

First o f all , Rokhli n wa s a geomete r an d topologis t wit h a specia l tast e fo r algebraic methods. Hi s knowledge and understanding o f various mathematical fields allowed hi m t o se e deep interconnection s an d th e unit y o f mathematics . Th e wor k of Rokhli n i s characterize d b y vas t mathematica l culture , a sens e o f th e new , an d an exquisit e writing style . H e always tried t o pas s these qualitie s on to hi s student s and younge r colleagues .

During th e las t 2 5 year s o f hi s life , V . A . Rokhli n worke d a t Leningra d Uni -versity, wher e h e create d a stron g schoo l o f topology , rea l algebrai c geometr y an d dynamical systems . A numbe r o f hi s student s no w wor k a t leadin g universitie s o f Europe an d America . Thei r paper s ar e i n divers e mathematica l fields, ofte n fa r removed fro m Rokhlin' s ow n interests . Amon g hi s student s an d mathematician s personally influenced b y him, let us mention L. Abramov, R . Belinskaya, Ya. Eliash-berg, S . Finashin, M . Gromov, N . Ivanov, V. Kharlamov, N . Netsvetaev, V . Turaev , A. Vershik , O . Viro , S . Yuzvinsky, an d A . Zhubr .

A se t o f importan t biographi c materials , photo s an d recollection s devote d t o Rokhlin appeare d i n 199 9 in a volume o f his selecte d papers , edite d b y A . Vershik , with contribution s b y V . Arnold , S . Novikov , Ya . Sinai , an d A . Vershi k (MCCM E Publishers, Moscow , 1999) . I n thi s volume w e reproduce Rokhlin' s biograph y writ -ten b y on e o f th e editors . I n Augus t 199 9 a t th e Eule r Institut e i n St . Petersbur g the conference "Topolog y an d Dynamics : Rokhlin' s Memorial " wit h mor e than 10 0 participants fro m man y countrie s too k place . Proceeding s o f thi s conferenc e wil l appear i n Zapisk i Nauchny h Seminaro v POMI , 266 , 26 7 (Englis h translatio n i n

ix

x F O R E W O R D

J. Math . Sci.) . Th e issue s o f St . Petersbur g Math . J . 11 , no. 5 , an d Proceeding s of St . Petersbur g Math . Soc . 7 als o were devote d t o V . A . Rokhlin .

Here we would lik e to say a few words abou t Rokhlin' s life . H e had t o confron t the mos t seriou s hardship s an d t o suffe r fro m bot h th e Naz i an d th e Communis t systems. Hi s father wa s arrested an d sho t i n the perio d o f Stalinist purges . Durin g World War I I Rokhlin was incarcerated i n a German concentratio n cam p for almos t three years . After th e en d o f the wa r h e was sen t t o a Sovie t deportatio n cam p fo r former prisoner s o f war . Eve n durin g th e relativel y prosperou s perio d o f hi s sta y in Leningrad , Rokhli n ha d t o suffe r fro m permanen t clashe s wit h hostil e scientifi c and part y bureaucracy , whic h finally force d hi m t o retir e a t th e ag e o f 60.

Soviet authoritie s neve r allowe d Rokhli n t o tak e par t i n scientifi c conference s abroad (no t eve n i n th e countrie s belongin g t o th e so-calle d "socialis t camp") ; h e was prohibite d fro m goin g outsid e th e USSR . Rokhli n di d no t liv e t o se e the tim e when h e would b e abl e t o accep t invitation s fro m foreig n universities .

The influence o f Rokhlin on the Leningrad mathematica l communit y an d on the mathematics departmen t o f the universit y was very strong despite al l the obstacles . He initiate d th e first (i n Russia ) obligator y cours e i n topolog y fo r undergraduat e students. H e was a n excellen t lecture r an d a n admirabl e scientifi c adviser .

In th e presen t volum e w e have collecte d severa l paper s b y Rokhlin' s students , followers, an d thei r students . Th e topic s covere d b y thes e paper s reflec t th e wid e spectrum o f Rokhlin' s interest s (topology , geometry , rea l algebrai c geometry , dy -namical systems ) a s well as the evolution o f his students sinc e his death. W e do no t analyze th e conten t o f the paper s here , leaving i t t o the reade r t o find th e topic s of his/her interest .

We sincerel y hop e tha t thi s volum e wil l contribut e t o th e memor y o f thi s re -markable scholar , t o who m w e both ow e so much .

Vladimir Turaev , Anatol y Vershi k