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