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
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