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).
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