Meeting February

Simplicial Complexes

Topics simplices simplicial complexes abstract simplicial complexes geometric realizations nerves

If dim then is called an face Simplices Points edges triangles and tetrahedra

and are improper faces and all others are proper are lowdimensional examples of simplices We use com

faces of The numb er of faces of is equal to the binations of p oints to dene simplices in general dimen

numb er of ways we can cho ose from k p oints sions Let S b e a nite set in ddimensional Euclidean

d

which is space denoted as R An ane combination of the

P P

p oints p S is a p oint x p with

i i i i

k k

The ane hul l a S is the set of all ane combina

k

tions Equivalently it is the intersection of all hyp er

planes that contain S The p oints in S are anely

The total numb er of faces is

independent of a i if none is the ane combination

k

of the other p oints in S For example the ane hull of

X

k

k +1

three a i p oints is a plane that of two a i p oints is

=1

a line and the ane hull of a single p oint is the p oint

itself

Simplicial complexes A simplicial complex is a

A convex combination is an ane combination with

nite collection K of simplices such that

nonnegative co ecients for all p S The

i i

convex hul l conv S is the set of all convex combina

tions Equivalently it is the intersection of all half

i K and implies K and

spaces that contain S A simplex is the convex hull of

d

ii K implies

a set of a i p oints If S R is a set of k a i

p oints then the dimension of the simplex conv S

Note that is a face of every simplex and thus b elongs

is dim k and is called a k simplex The largest

d

to K by condition i Condition ii therefore allows

numb er of a i p oints in R is d and we have sim

for the p ossibility that and b e disjoint Figure

plices of dimension through d Figure shows the

3

shows three sets of simplices that violate one of the

ve typ es of simplices in R The p oints in any sub

two conditions and therefore do not form complexes

A subcomplex is a subset that is a simplicial complex

0

-1 0 1 2 3

Figure The simplex is the empty set a simplex

Figure To the left we are missing an edge and two

is a p oint or vertex a simplex is an edge a simplex is

vertices In the middle the triangles meet along a segment

a triangle and a simplex is a tetrahderon

that is not an edge of either triangle To the right the

edge crosses the triangle at an interior p oint

set T S are a i so the convex hull of T is again a

simplex Sp ecically conv T is the subset of p oints

x with whenever p S is not in T The itself Observe that every subset of a simplicial complex

i i

simplex is a face of and we denote this relation by satises condition ii To enforce condition i we add

faces and simplices to the subset Formally the closure concepts of face sub complex closure star link extend

of a subset L K is the smallest sub complex that straightforwardly from geometric to abstract simplicial

contains L complexes

The set system together with the inclusion relation

Cl L f K j Lg

forms a partially ordered set or p oset denoted as

A Posets are commonly drawn using Hasse di

A particular sub complex is the iskeleton that consists

agrams where sets are no des smaller sets are b elow

of all simplices K whose dimension is i or less The

larger sets and inclusions are edges see Figure In

vertex set is

clusions implied by others are usually not drawn To

Vert K f K j dim g

it is the same as the skeleton except it do es not con

tain the simplex The dimension of K is the

largest dimension of any simplex dim K maxfdim j

K g If k dim K then K is a k complex and it is

pure if every simplex is a face of a k simplex The un

derlying space is the set of p oints covered by simplices

S S

j K j K A polyhedron is the underlying

2K

Figure From left to right the p oset of the empty set a

space of a simplicial complex Sometimes it is useful

vertex an edge a triangle a tetrahedron

draw the Hasse diagram of a k simplex we draw the

Hasse diagrams for two k simplices One is the

diagram of a k face of and the other is the

diagram for the star of the vertex u Finally

we connect every simplex in the star of u with the

simplex fug in the closure of An abstract sim

Figure Star and link of a vertex To the left the solid

plicial complex A is a subsystem of the p ower set of

edges and shaded triangles b elong to the star of the solid

Vert A We can therefore think of it as a sub complex of

vertex To the right the solid edges and vertices b elong to

an nsimplex where n card Vert A This view is

the link of the hollow vertex

expressed in the picture of an abstract simplicial com

plex shown as Figure

to consider substructures of a simplicial complex The

star of a simplex consists of all simplices that contain

and the link consists of all faces of simplices in the

Vert A

star that do not intersect

St f K j g

f Cl St j g Lk A

0

see Figure for examples The star is generally not

closed but the link is always a simplicial complex

Figure The onion is the p ower set of Vert A The area

b elow the waterline is an abstract simplicial complex

Abstract simplicial complex If we replace each

simplex in a simplicial complex by the set of its ver

tices we get a system of subsets of the vertex set In

Geometric realization We can think of an abstract

doing so we throw away the geometry of the simplices

simplicial complex as an abstract version of a geometric

which allows us to fo cus on the combinatorial struc

simplicial complex To formalize this idea we dene a

ture Formally a nite system A of nite sets is an

geometric realization of an abstract simplicial complex

abstract simplicial complex if A and implies

A as a simplicial complex K together with a bijection

A This requirement is similar to condition i for

Vert A Vert K such that A i conv K

geometric simplicial complexes A set A is an ab

Conversely A is called an abstraction of K

stract simplex and its dimension is dim card

S S

Given A we can ask for the lowest dimension that The vertex set of A is Vert A A The

2A

allows a geometric realization For example graphs are region to its generator V a This function de

a

dimensional abstract simplicial complexes and can al nes a geometric realization of Nrv C

3

ways b e realized in R Two dimensions are sometimes

D fconv j Nrv C g

but not always sucient This result generalized to

k dimensional abstract simplicial complexes they can

This is the Delaunay triangulation of S What hap

2k +1 2k

always b e realized in R and sometimes R do es not

p ens if the p oints in S are not in general p osition If

suce To prove the suciency of the claim we show

k Voronoi regions have a nonempty common

that the k skeleton of every nsimplex can b e realized in

intersection then Nrv C contains the corresp onding ab

2k +1

R Map the n vertices to p oints in general p o

stract k simplex So instead of making a choice among

2k +1

sition in R Sp ecically we require that any k

the p ossible triangulations of the k gon the nerve

of the p oints are a i Two simplices and of the k

takes all p ossible triangulations together and interprets

skeleton have a total of at most k vertices which

them as sub complexes of a k simplex The disadvan

are therefore a i In other words and are faces of a

tage of this metho d is of course the fact that Nrv C can

common simplex of dimension at most k It follows

2

no longer b e realized in R

that is a common face of b oth

Theorem Every k dimensional abstract simplicial

Bibliographic notes During the rst half of the

2k +1

complex has a geometric realization in R

twentieth century combinatorial top ology was a our

ishing eld of Mathematics We refer to Paul Alexan

drov as a comprehensive text originally published as

Nerves A convenient way to construct abstract sim

a series of three b o oks This text roughly coincides with

plicial complexes uses arbitrary nite sets The nerve

a fundamental reorganization of the eld triggered by a

of such a set C is the system of subsets with nonempty

variety of technical results in top ology One of the suc

intersection

cessors of combinatorial top ology is mo dern algebraic

top ology where the emphasis shifts from combinatorial

Nrv C fX C j X g

to algebraic structures We refer to James Munkres

T T

for an intro ductory text in that area

If Y X then X Y Hence X Nrv C im

plies Y Nrv C which shows that the nerve is an

We proved that every k complex can b e geometri

2k +1

abstract simplicial complex Consider for example the

cally realized in R Examples of k complexes that

case where C is a covering of some geometric space such

require k dimensions are provided by Flores and

as in Figure Every set in the covering corresp onds

indep endently by van Kamp en One such example

is the k skeleton of the k simplex For k

this is the complete graph of ve vertices which is one

of the two obstructions of graph planarity identied by

Kuratowski

P S Alexandrov Combinatorial Topology Dover

New York

A Flores Ub er ndimensional e Komplexe die in

Figure A covering with eight sets to the left and its

R selbstverschlungen sind Ergeb Math Kol l 6

2n+1

nerve to the right The sets meet in triplets but not in

quadruplets which implies that the nerve is dimensional

E R van Kampen Komplexe in euklidisch en Raumen

Abh Math Sem Univ Hamburg 9

to a vertex and k sets with nonempty intersection

dene a k simplex

K Kuratowski Sur le probleme des courb es en top olo

gie Fund Math 15

We have seen an example of such a construction ear

2

lier The Voronoi regions of a nite set S R dene

J R Munkres Elements of Algebraic Topology Addi

a covering C fV j a S g of the plane Assuming

sonWesley Redwo o d City California

a

general p osition the Voronoi regions meet in pairs and

in triplets but not in quadruplets The nerve therefore

consists only of abstract vertices edges and triangles

2

Consider the function C R that maps a Voronoi