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.
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