A TOPOLOGICAL CHERN-WEIL THEORY 9

9ZA

2^1

Figure 1.2: The standard subdivision of

C2

and

C3

A

a

( C r ) = {($!, . . . , Sr)|sa(i) 5a(2) - . . 50(r)}*

The vertices of

Acr(Crr)

are

Vaj = ( 5 «(1) = • • • = 3

a

(j) = 1; S

a

(j+l) = • • • = S

a

(

r

) = 0)

for j = 0 , . . . , r. With respect to these vertices, the barycentric coordi-

nates are

ta,Q = 1 — Sa(l)

W = Sa(i) - 3a(2)

*cr,r-l = 5

a

(

r

«i ) - S

a

(

r

)

The A

a

( C

r

) and their faces, for all permutations a, form a simplicial

complex, subdividing C

r

, which we will denote by A(C

r

) , the standard

subdivision of C

r

. As oriented chains,

Cr

= J2a

e

(

a

) A

a

( C

r

) , where e(a)

is the sign of the permutation a.

1.5 Standard subdivision of a product of simplexes. Let e, be the i-

th standard basis vector in R

n

, numbered starting with i = 0. Let A* =

e

0

,...,efc C R*

+1

, A ' = e

0

, . . . , e * C R'

+ 1

. Let / = {*i,...,«*}, 1

ii ii • • • ik k + £, be any fc-subset of {1, ...,& + £}y and let

J = {ii? • • • j*} 1 ji ]2"' je fc + £ be the complemen-

tary subset. We take the (ife + ^-simplex A/* ' C A* x A ' C

R*+'+2