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")