Introduction

In these notes we present th e first steps of a theory of confoliations whic h is designed

to lin k th e geometr y an d topolog y o f 3-dimensiona l contac t structure s wit h th e

geometry an d topolog y o f codimensio n 1 foliations o n 3-manifolds . A confoliatio n

is a mixe d structur e whic h interpolate s betwee n a codimensio n 1 foliatio n an d

a contac t structure . Thi s objec t (withou t th e name ) first appeare d i n wor k o f

Steve Altschule r (se e [2]) . I n thi s pape r w e will b e concerne d exclusivel y wit h th e

3-dimensional case , althoug h confoliation s ca n b e define d o n highe r dimensiona l

manifolds a s well (se e Sectio n 1.1. 6 below) .

Foliations an d contac t structure s hav e bee n studie d practicall y independently .

Indeed a t th e firs t glance , these object s belon g to two different worlds . Th e theor y

of foliations i s a part o f topology an d dynamica l system s while contact geometr y i s

an odd-dimensional brother of symplectic geometry. However , the two theories have

developed a numbe r o f strikin g similarities . I n eac h case , a n understandin g devel -

oped that additiona l restriction s ar e important o n foliations an d contac t structure s

to mak e the m interestin g an d usefu l fo r applications , fo r otherwise , thes e struc -

tures ar e s o flexible tha t the y fi t anywhere , henc e produc e n o informatio n abou t

the topolog y o f th e underlyin g manifold . Thi s extr a flexibility i s cause d b y th e

appearance o f Reeb components i n th e cas e o f foliation s (se e [47] , [55] , [53]) , an d

by th e overtwisting phenomenon i n th e cas e o f contac t structure s (se e [6] , [10]).

In orde r t o mak e th e structure s mor e rigi d i n th e contex t o f foliation s th e theor y

of taut foliations wa s develope d ([52] , [51] [19]), a s wel l a s th e relate d theor y o f

essential lamination s ([20] , [31]). I n th e paralle l worl d o f contac t geometry , th e

theory o f tight contact structures wer e develope d fo r simila r purpose s ([6] , [11],

[23]).

The theor y o f confoliations shoul d hel p u s t o bette r understan d link s betwee n

the two theories and shoul d provid e an instrument fo r transporting th e results fro m

one fiel d t o th e other .

Acknowledgements. Th e authors benefited a lot from discussion s with man y

specialists i n bot h fields. W e ar e especiall y gratefu l t o E . Ghys , E . Giroux , A .

Hatcher an d I . Katznelson . E . Ghy s prove d Theore m 1.2.7 a s a n answe r t o ou r

question. W e thank R. Hind for attentive reading of the manuscript an d A. Haeflige r

for man y critica l comments , usefu l suggestion s an d fo r providin g u s wit h severa l

references.

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