A TOPOLOGICAL CHERN-WEIL THEORY 11

1.7 The geometric bar construction. Let X

0

,... ,X

r

be any com-

pact topological spaces. We define

X

0

[XL|

• • • \Xr], the geometric bar

product, by induction on r.

X0[ ] is just X0; the brackets are included only to simplify certain

formulas.

If XJX2I • • • \Xr] has been defined, then

Xo[*i| • • • \Xr] = X0x cone(X1[X2| • • • \Xr]).

If Xo,..., Xr are all points, then the bar product Xo[Xi| • • • \Xr] is

just the simplex XQ, ... , Xr . Furthermore in general we have the

following "barycentric" coordinates on Xo[Xi| • • • \Xr]: it sits inside the

r-fold join

X0 * (XQ x Xi) * (X0 x

XI

x X2) * • • • * (X0 x • • • x Xr)

as the subset

{t0XQ + ti(x0,Xi) + t2(x0,Xi,X2) H \-tr(x0,Xi,. . . ,#

r

)}

where 0 t,- 1 and J2U = 1- It follows that for any r-fold bar

product r = Xo[Xi| • • • \Xr] there is a natural projection

7rr:

r — »

Ar

defined by

nr(t0xo

+ i(so, a?i) H h tr(x0, xu ..., xr)) = t0e0 + *iei + h i

r

e

r

.

1.8 We shall use the geometric bar construction only when the X{ are

simplices or cubes. In such cases, and indeed whenever the X{ are all

connected, oriented manifolds, we have

dimXo[Xi| • • • \Xr] = j^dimXj + r.

i=o

and

d(x0[x1|-HX-]) = d*o[Xi|---|X]

+ B-i)

r +

'

w

^

0

[^il-|ax

i

|...|jf

r

]

+(X0xX1)[X2\---\Xr]

+ Jbi-lYMXil • • • l*i-i|*i x ^i+il^i+al • • • M

+(-l)'-e(X

r

)Xo[X

1

|.-.|X

r

_

1

],

where S(j) = X^dimX,-, and e(X) = 1 if dimX = 0, and = 0

otherwise. (See Figure 1.3).