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
Previous Page Next Page