16 ANTHONY V. PHILLIPS and DAVID A. STONE
The counit is the augmentation map
, x f 0, dim
G
0
dim a = 0.
There is also a standard comultiplication [16, Equation 9.3.3] on
any locally ordered cubical complex K. Let C G K be an n-cube with
coordinates $ i , . . . , s
n
, in that order. For each partition II of { 1 , . . . , n}
into II' U II", let Fi\C and B\\C be the faces determined respectively
by the conditions st- = 0 for i G II" and S{ = 1 for i G II'. Say II' =
{z'i,... ,ip} and II" = {z",... , z"_
p
}, with the z' and z" listed in order,
and let e(II) be the sign of the permutation (i[,..., z^, z",..., z"_p) of
( 1 , . . . , n). Then V
K
is defined by
VK(C) = ^2e(Il)FnC®BnC.
n
L e m m a 1.13 Let K be a locally ordered cubical complex and A its
standard simplicial subdivision. Then A* ® A* is a subdivision of K% ®
/*; and V
A
corresponds to V
x
, in the following sense: let C G K and
let C = Yla
e(a)Aa
be its standard subdivision. Then the subdivision
of
VK(C)
in A, ® A, is E a e(a)V
A
(A
a
).
Proof. Let C be an n-cube, with coordinates s i , . . . , s
n
. For each
k = 0 , . . . , n, let P(fc) be the set of partitions II = ITU 11" of { 1 , . . . , n}
in which |H'| = fc, |II"| = n - k. Let 5(11'), S(II") denote respectively
the groups of permutations of the elements of
II7
and II". For each II
we have the standard subdivisions FuC = ]Ca'eS(rr)
e(a')Aai
B\\C =
£«"€«s(n") e(a")Aa", (where Aat = Aa(FnC), Aan = A
a
//(5
n
C) , in the
notation of 1.4).
Now
V*(C) = E
e
(
n
)
F
nC®£nC
n
E E n)( E *("OA«o®( E «(«")*«»)
A:=0 Tl€V(k) a'€«S(II') at"eS(Tl")
= £ £
c(n)c(a/)c(a//)Aa,®AaW.
A;=0 ne^(A:)
a'€5(n')
a"€5(n")
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