A TOPOLOGICAL CHERN-WEIL THEORY 15
and the pair
f (su---,3j-u8j = M j + i = ••• = 3
r
= 0) V(0,j)
\ (Si = = 3; = 1, 3
i +
i , . . . , 5
r
) £ Cjj(j)
Assume by induction that the lemma is true for each Cu(j), j =
1 , . . . , r. That is for each subset / ' = (j = i\ i2 - - - ip = r) set i,
st- = l,z j
Si sik^ for ik i ik+i,k = 2 , . . . ,p - 1
3,-j S;2
sip
and assume that IV = Vj[V^j, 22)! |V(i
p
_i,r)] (where V^ = (si =
•.. = sj = 1, Sj+i = = sr = 0)), that {IV and faces } form a cellular
subdivision T(Ca(j)) of Cu(j), and that A(Cu(j)) is a simplicial subdivi-
sion of r(Cu(jf)). Then
Cr
has a cellular decomposition
T"(Cr)
= {Vo*
(V(0, j) x IV) and their faces, j = 1 , . . . , r and / ' as above }. Here VJ *
(V(0, j ) xTp) on the one hand equals Vo[V(0J)\V(j,i2)\ |V(i
p
_i,r)]
and, on the other hand, is given by the inequalities
( Si Sj valid on all of Vo * d*(C)
Si Sik for ik-i i ik
[ Si2--- 3ip.
Thus V
o
*(^(0,j)xIV ) = T7, where / = (0 = iQ,j = ix i2
ip = r). Conversely, given any / = (0 = %Q i\ ip r), set
j = ti; then, w i t h / ' a s above, Tj = Vo*(V(0J) x I V ) . Now(l) and (2)
follow; and (3) follows from the fact that A(C
r
) =
Urj=1{V0
* A(Cjj(j))
and its faces } .
1.12 Comultiplications . A comultiplication on a chain complex /C*
is an algebraic model for the diagonal map X —• X x X; it is a chain
map V^: /C* /C* ® /C* such that:
1) (V* ® lJV * = (1 ® V
; c
)V
/ c
as maps /C* - £* ® /C* ® £*;
2) there is a counit e: /C* - Z such that (1 ® eJV* = (c ® 1) V * =
id/c„. The triple (/C*, V ^ e ) is then called a coalgebra.
Any locally ordered simplicial complex (A, o) has a standard comul-
tiplication ([20, page 45], [16, Prop. 9.2.5]), given by
dim
a
j=0
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