QUANTITATIVE THEOR Y O F FOLIATION S
From thi s las t definitio n w e se e tha t a foliatio n ca n b e considere d a s a generaliz
differentiable structur e o n A t (Differentiabi e structure s ar e jus t foliation s o f dimensi
zero.)
For ou r nex t remar k w e nee d th e followin g observations .
(1.3)
(1.4)
^ a = ^ 8 a onf a{UxOUx,).
0 * = £* £ x wheneve r th e equatio n make s sense .
These statement s ca n b e interprete d a s follows . Le t T r denot e th e se t o f germ s o f l
cal C r diffeomorphism s o f R* . Thi s se t is give n th e natura l "sheaf " topolog y havin
a basi s consistin g o f set s o f th e for m !(£ ) : x domain(j6)} wher e 0 i s a loca l C r
diffeomorphism o f R * an d (f) . denote s th e ger m o f c/ a t x. (Thus , a sequenc e y
y i n r ^ i f an d onl y i f ther e i s a loca l C T diffeomorphis m f o f R ^ an d a sequenc
points x. x i n domain(£ ) suc h tha t y . = (f)
x
. for / ' sufficientl y large. ) Th e map
a, r: T r * R^, whic h assig n t o a give n ger m it s sourc e an d targe t point s respectively
are loca l homeomorphisms . Wheneve r y , y ' Tr satisf y th e conditio n oiy') = r(y) , th
composition y' ° y i s defined . Thi s pairin g i s continuou s o n th e subse t o f composabl
pairs i n T r x T . Th e unit s i n T r ar e th e germ s o f th e identit y transformatio n a t poi
of R^ , denote d 1 fo r x 6 R*. The y for m a subse t homeomorphi c (vi a eithe r a o r r)
R^. Eac h y £Yg ha s a n "inverse " y suc h tha t y~ l o y ^ 1 . I n brief , T r i s a
so-called topological groupoid (cf . Chapte r IV) .
Statement (1.3) ca n no w b e interprete d a s sayin g tha t th e assignmen t x * )
from Definitio n l.l " i s a continuou s ma p 0 o
a
: U
a
O 'UQ~* Vr fo r eac h a , j8 . Stateme
(1.4) the n say s tha t i n eac h U
a
C\ Uo O U , thes e map s satisf y th e cocycl e condition
In thi s form , a foliatio n ha s th e structur e o f a principa l bundl e ove r M wher e th e
fiber, instea d o f bein g a topologica l group , i s a topologica l groupoid . Thi s poin t o f vi
is valuable , fo r i t turn s ou t tha t muc h o f th e algebrai c topologica l machiner y develope
to stud y principa l G-bundle s wher e G i s a topologica l grou p ca n b e generalize d t o th
case wher e G i s a topologica l groupoid . Fo r example , on e ca n defin e classifyin g spac
and characteristi c classe s fo r foliations . Furthermore , usin g thi s machinery , on e ca n
give a homotopy-theoreti c classificatio n o f th e foliation s (u p t o a certai n equivalence
on an y give n manifold . I n particular , i t i s ofte n possibl e t o decid e whethe r o r no t a
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