10 ANTHONY V. PHILLIPS and DAVID A. STONE

to be the convex span of the vertices uo,... ,Vk+t, where, inductively,

v0 = (e

0

,e

0

); and if v{ = (e

p

,e

g

), then

„ _ J (*p+i,eg) if i + l e l

+ 1

I (e

P

,e

g + 1

) if i + leJ

Then the

A*+*

and their faces form a simplicial subdivision of

Ak

x A^

which we will call the standard subdivision. The orientation of

A*+*

is the sign of the permutation i

1 ?

.. . ,ifc,ji,... , j ^ . This subdivision

process is associative.

(A graphic way to represent this process is to place the vertices of

A* X A* in a rectangular array where the columns are labelled from

left to right eo,.. . , e^, and the rows are labelled from bottom to top

eo,.. . ,ejfc. Then each subset I as above corresponds to & path from

the lower left-hand corner to the upper right; the path consists of k + £

steps taken according to the rule: at the i-th step move up if i G / , and

move to the right otherwise.)

1.6 Orderings. A simplicial complex A is locally ordered by giving a

partial ordering o of its vertices in which the vertices of each simplex are

totally ordered. We shall usually refer to "the locally ordered simplicial

complex A" and mention o only when necessary. For example, the first

barycentric subdivision of any simplicial complex has a natural local

ordering: if f and a are barycenters of r and a respectively, then f - a

if r is a face of a. A local ordering o of a finite set can be refined to a

total ordering: start with any o-minimal vertex; and inductively select

any vertex which is o-minimal among those not yet selected.

An ordering of the vertices of an r-simplex a determines a unique

order-respecting linear homeomorphism a — * A

r

and picks out faces

dja,

CTU(Z)

and crdj) corresponding to the similarly labelled faces of A

r

.

A cubical complex K is locally ordered when on each cube C are

specified coordinates in order, s f , . . . , $ £ (where n — dim C), such

that if

C1

is either d{C or

dlC

then under the face map

Cf c-

C

the order of the coordinates $?' agrees with that of s^,..., sf,..., s%.

Any locally ordered cubical complex has a (locally ordered) standard

simplicial subdivision.