A TOPOLOGICAL CHERN-WEIL THEORY 13

1.9 R e m a r k , The geometric bar construction is analogous to the

(algebraic) inhomogeneous unreduced bar construction, defined for any

differential graded augmented algebra A* as follows: B* = B(A*, A*, R )

is defined to be ©

r 0

®

r + 1

A*; where an element of

g)r+1

A* is usually

written Xo[Xi| • • • \Xr], X{ £ A*. The grading and the boundary op-

erator on B+ are given by the same formulas as those above. (See

2.6 and McCleary's book [21]).Our sign conventions agree with those

of MacLane [20] and Milgram [23]; the notations

J B ( , 4 * , . 4 * , R )

and

J3(R, A*,R) are as in McCleary.

1.10 Subdivision of

CT

into b a r - p r o d u c t s of cubes . We shall

now describe a subdivision of the cube

Cr

into bar-products of cubes.

Whenever 0 i j r, consider the set CA(I) D C^(j) C C

r

, given

in coordinates by Sk = 1, k i, and Sk = 0, k j ; this set is naturally

parametrized by Sj+i? ...,Sj. Set

v(i,j) = &cfi)ncb(j).

Now let I be any subset of { 0 , . . . , r} containing 0 and r, say I =

{z'o, h , . . . , ip} with (0 = z'o h • • • ip = r). Let Tj be the subset

of

Cr

given by the inequalities

Si 5ifc+1 for ik i ik+u fc = 1,"... ,p - 1

L e m m a 1.11 (See Figure 1.4). (1) T/ is isomorphic to the geometric

bar-product Vo[V{0,ii)\V(iui2)\-'\V{i^ur)].

(2) As / varies, the T/ and their faces form a cellular decomposition

T(Cr)

of the cube

Cr.

Set 0j = XX=i(u — A:). Then, as oriented r-

chains, C

r

= E / ( - l ) *

7

r / .

(3) The standard (simplicial) subdivision

A(Cr)

is a subdivision of

T(C

r

).

Proof. We regard

Cr

as the cone from its vertex VQ (where all S{ = 0)

on

d+Cr

= Uj=i 5

j

C

rr

. Each

^'Cr

can be identified with V(0, j ) x C^j)

by the correspondence between

( « ! , . . . , ^ j - i ^ j = l , 5

j +

i , . . . , 5

r

) G

dJCr