Motives and oriented cohomology of generically cellular varieties

Abstract: For a cellular variety

X

over a field

k

of characteristic 0 and an algebraic oriented cohomology theory

h h

of Levine-Morel we construct a filtration on the cohomology ring

h h (X)

such that the associated graded ring is isomorphic to the Chow ring of

X

. Taking

X

to be the variety of Borel subgroups of a split semisimple linear algebraic group

G

over

k

we apply this filtration to relate the oriented cohomology of

G

to its Chow ring. As an immediate application we compute the algebraic cobordism ring of a group of type

G_{2}

, of groups

S O_{n}

and

S p i n_{m}

for

n = 3, 4

and

m = 3, 4, 5, 6

and

P G L_{k}

for

k g e q s l a n t 2

. Using this filtration we also establish the following comparison result between Chow motives and

h h

-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically split variety

Y

gives rise to a

h h

-motivic decomposition of

Y

with the same generating function. Moreover, under some conditions on the coefficient ring of

h h

the obtained

h h

-motivic decomposition will be irreducible. We also prove that if Chow motives of two twisted forms of

Y

coincide, then their

h h

-motives coincide as well.