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 .