Computational Topology (Jeff Erickson)
Cell Complexes: Definitions
One who does not realize his own value is condemned to utter failure.
(Every kind of complex, superiority or inferiority, is harmful to man).
— ‘Al¯
ı ibn Ab¯
ı T
.
¯alib,
Nahj alBalagha [Peak of Eloquence]
There once lived a man named Oedipus Rex.
You may have heard about his odd complex.
His name appears in Freud’s index
’Cause he loved his mother.
— Tom Lehrer, “Oedipus Rex”,
More of Tom Lehrer
(1959)
15
Cell Complexes: Definitions
With a few exceptions, we have focused exclusively on problems involving 2manifolds, represented by
polygonal schemata. Before we (finally!) consider problems involving more general spaces, we need to
step back and consider how these more general spaces might be represented, as part of the input to an
algorithm. This is a surprisingly subtle question, even if we restrict our attention to higherdimensional
manifolds
.
A large class of topological spaces of practical interest can be represented by a decomposition into
subsets, each with simple topology, glued together ‘nicely’ along their boundaries. A decomposition of
this form is commonly called a
cell complex
. Abstract graphs (as branched 1manifolds) are examples of
cell complexes, as are polygonal schemata, quadrilateral and hexahedral meshes, and 3d triangulations
supporting normal surfaces. There are several different ways to formalize the intuitive notion of a ‘cell
complex’, striking different balances between simplicity and generality. I’ll describe several different
definitions in this lecture. The figure below shows the ‘cell complex’ of minimum complexity (number of
cells) homeomorphic to the torus, for four standard definitions.
The simplest CW complex,
Δ
complex, regular
Δ
complex, and simplicial complex homeomorphic to the torus.
15.1
Abstract Simplicial Complexes
An
abstract simplicial complex
is a collection
X
of finite sets that is closed under taking subsets; that is,
for any set
S
∈
X
and any subset
T
⊆
S
, we have
T
∈
X
. In particular, the intersection of any two sets
in
X
is also a set in
X
, and the empty set
∅
is an element of every simplicial complex.
The sets in
X
are called
simplices
. The
vertices
of
X
are the singleton sets in
X
(or their elements);
the set of vertices of
X
is denoted
X
0
. The
dimension
of a simplex is one less than its cardinality; in
particular, every vertex is a 0dimensional simplex, and the empty set has dimension

1. The
dimension
of a simplicial complex is the largest dimension of any simplex. The subsets of a simplex are called
its
faces
. A simplex in
X
is
maximal
if it is not a proper face of any other simplex in
X
; the maximal
simplices in
X
are also called the
facets
of
X
. An abstract simplicial complex
X
is
pure
if every facet has
the same dimension.
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 Fall '08
 Staff
 Algebraic Topology, Polytope, Simplicial complex, Simplicial complexes, Geometric Simplicial Complexes

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