A TOPOLOGICAL CHERN-WEIL THEORY 17
For fixed A;, there is a one-to-one correspondence between the set
of permutations a of { 1 , . . . ,n} and the set of triples (II, a', a") with
II G
V(k),ctl
e S(W) and a" G S(II"). The correspondence is de-
fined in the following way. Given a, set II' = {c*(l),... , a(&)}, II" =
{a(fc+l),... , #(n)}. Say the elements of II' and II", when listed in their
natural (increasing) orders, are (respectively) i'ly..., i'k and i",... , z"_*.;
and let a\ a" be the permutations of these orderings to the orderings
a ( l ) , . . . , a(k) and a(k + 1 ) , . . . , a(n) respectively. This defines a func-
tion a (II, a', a"). Conversely, given (II, a', a"), set
{P)
I « * W ^ P *
This gives a function (II, a', a") a, inverse to the previous one.
Moreover, since a is the composite of the permutations ( 1 , . . . ,n) —•
(.V..,«,...,C*) - (o/Ci;),...,^.;),«"(»•?),.-..""(C*)). the
sign of a is e(a) = = e(n)e(a')e(a"). Therefore
The vertices of A
a
are, in order, J?j=i
e«0)?
for z = 0 , . . . , n; that is,
e
0
, ea(!) , e
a(1 )
+ ea(2) , . . . , £*
= 1
**{$) = ( 1 , 1 , . . . , 1). The vertices of
A
a
/ in FY\C have coordinates corresponding to indices in 11" equal to 0;
so these vertices are, in order, J2)=i ea'(j)5 for z = 0,...,fc = dim FnC
The vertices of Aa» in BnC have coordinates corresponding to indices
in IT equal to 1; so these vertices are ]Cj=i
ea'(j)
+ Ej=i
ea"(j)i

r
z = 0 , . . . , n k = dim B\\C. Thus A
a
/ = A u(fc), A
a
" = A u(fc) when
a is related to oi\ a" as above. So
V
C
(C ) = f(a)Aab(A:)gAQJj(fc)
fc=0
= V
A

a
e(ct)A
a
) .
Thus the lemma holds on each cube of K, and the lemma itself
follows.
The following lemma is standard (see [20]).
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