Chapter 1
Combinatorial preliminaries
This chapter may be skipped at first reading and referred back to when
necessary.
1.1 Let
Cr
be the standard r-cube in R
r
, that is,
Cr
= {(si,.. . , s
r
)|
0 S{ 1}. For j = 1,.. . ,r the faces d*Cr and djCr are defined by
the equations Sj = 1 and Sj = 0 respectively. Set 9+ =
£(—l)J'+1dJ'
and 9 + = X3(—iV^jJ then, with the standard orientations on
Cr
and
C
r
- \
9Cr
= d+C
r
+ d+C
r
. Moreover,
92
=
(d+)2
=
(9+)2
= 0.
Let CMi) and CHj) be the back (r i)-face and the front j face of
C
r
, respectively, for 0 i,j r. Cji(i) is defined by Sk = 1, fc z and
CJ(j) by a* = 0, fc j . In particular, CJ(O) = CjJ(r) = C
r
.
1.2 Let A
r
be the standard r-simplex in R
r
, that is, A
r
= {toeo+- •+
trer} where eo is the origin ( 0 , . . . , 0), et is the ith. standard basis vector
and t0 + + tr = 1. We call (t0,..., tT) the barycentric coordinates of
t0e0-i trer. More generally, we use letters v, w,.. to denote vertices of
a simplicial complex A, and the notation a = i?o,..., vr to indicate
that 7 is a simplex of A with those vertices. In contexts where a is being
considered in isolation from A, we often write simply a = 0 , . . . , r .
Let
djAr
= e o , . . . , e j , . . . , e
r
, j = 0 , . . . , r, and let Ajj(i) and
Ar(i) be the back (r i)-face and the front j-face of A
r
, respectively:
Aj(i) = e
t
, . . . , e
r
and AJ(jf) = e0, . . . ^ .
7
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