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IAS Park City Mathematics Series,Volume 14 2004,Topics in Combinatorial Di erential. Topology and Geometry,Robin Forman, Many questions from a variety of areas of mathematics lead one to the problem. of analyzing the topology or the combinatorial geometry of a simplicial complex. We will see a number of examples in these notes Some very general theories have. been developed for the investigation of similar questions for smooth manifolds Our. goal in these lectures is to show that there is much to be gained in the world of. combinatorics from borrowing questions tools motivation and even inspiration. from the theory of smooth manifolds, These lectures center on two main topics which illustrate the dramatic impact. that ideas from the study of smooth manifolds have had on the study of combina. torial spaces The rst topic has its origins in di erential topology and the second. in di erential geometry, One of the most powerful and useful tools in the study of the topology of smooth. manifolds is Morse theory In the rst three lectures we present a combinatorial. Morse theory that posesses many of the desirable properties of the smooth theory. and which can be usefully applied to the study of very general combinatorial spaces. In the rst two lectures we present the basic theory along with numerous examples. In the third lecture we show that discrete Morse theory is a very natural tool for. the study of some questions in complexity theory, Much of the study of global di erential geometry is concerned with the rela.

tionship between the geometry of a Riemannian manifold and its topology One. long conjectured still unproved relationship is the Hopf conjecture which states. that if a manifold has nonpositive sectional curvature then the sign of its Euler. characteristic depends only on its dimension See Lecture 4 for a more precise. statement In 15 Charney and Davis formulated a combinatorial analogue of. 1 The Department of Mathematics Rice University Houston TX USA 77251. E mail address forman rice edu, The author was supported in part by the National Science Foundation The author would also. like to thank Carsten Lange who served as the TA for these lectures created most of the gures. in these notes and whose enthusiasm and attention to detail dramatically increased the com. prehensibility of the text The author expresses his gratitude to the organizers of the IAS PC. Summer Institute for their tireless dedication and enthusiasm for all things organizational and. mathematical Their support greatly improved the lectures and these notes. 2007 American Mathematical Society, 136 R FORMAN COMB DIFFERENTIAL TOPOLOGY AND GEOMETRY. this conjecture and then observed that their conjecture is related to some of the. central questions in geometric combinatorics There has been some fascinating re. cent work on this subject which has resulted in some very tantalizing more general. conjectures In Lectures 4 and 5 we present an introduction to the conjectures of. Charney and Davis discuss some of the known partial results and survey the most. recent developments,Discrete Morse Theory,1 Introduction. There is a very close relationship between the topology of a smooth manifold M. and the critical points of a smooth function f on M For example if M is compact. then f must achieve a maximum and a minimum Morse theory is a far reaching. extension of this fact Milnor s beautiful book 71 is the standard reference on. this subject In these notes we present an adaptation of Morse theory that may be. applied to any simplicial complex or more general cell complex There have been. other adaptations of Morse Theory that can be applied to combinatorial spaces For. example a Morse Theory of piecewise linear functions appears in 59 and the very. powerful Strati ed Morse Theory was developed by Goresky and MacPherson. 46 47 These theories especially the latter have each been successfully applied. to prove some very striking results, We take a slightly di erent approach than that taken in these references Rather. than choosing a suitable class of continuous functions on our spaces to play the role. of Morse functions we will be working entirely with discrete structures Hence we. have chosen the name discrete Morse theory for the ideas we will describe More. over in these notes we will describe the theory entirely in terms of the discrete. gradient vector eld rather than an underlying function We show that even with. out introducing any continuity one can recreate in the category of combinatorial. spaces a complete theory that captures many of the intricacies of the smooth the. ory and can be used as an e ective tool for a wide variety of combinatorial and. topological problems, The goal of these lectures is to present an overview of the subject of discrete.

Morse theory that is su cient both to understand the major applications of the. theory to combinatorics and to apply the the theory to new problems We will. not be presenting theorems in their most recent or most general form and simple. examples will often take the place of proofs Those interested in a more complete. presentation of the theory can consult the reference 32 Earlier surveys of this. work have appeared in 31 and 35 and earlier and similar versions of some of. the sections in these notes appeared in 39 and 40, 138 R FORMAN COMB DIFFERENTIAL TOPOLOGY AND GEOMETRY. 2 Cell Complexes and CW Complexes, The main theorems of discrete and smooth Morse theory are best stated in the. language of CW complexes so we begin with an overview of the basics of such. complexes J H C Whitehead introduced CW complexes in his foundational work. on homotopy theory and all of the results in this section are due to him The. reader should consult 68 for a very complete introduction to this topic In these. notes we will consider only nite CW complexes so many of the subtleties of the. subject will not appear, The building blocks of cell complexes are cells Let B d denote the closed unit. ball in d dimensional Euclidean space That is B d x Ed s t x 1 The. boundary of B d is the unit d 1 sphere S d 1 x Ed s t x 1 A d cell. is a topological space which is homeomorphic to B d If is d cell then we denote. by the subset of corresponding to S d 1 B d under any homeomorphism. between B d and A cell is a topological space which is a d cell for some d. The basic operation of cell complexes is the notion of attaching a cell Let X. be a topological space a d cell and f X a continuous map We let X f. denote the disjoint union of X and quotiented out by the equivalence relation. that each point s is identi ed with f s X We refer to this operation by. saying that X f is the result of attaching the cell to X The map f is called. the attaching map, We emphasize that the attaching map must be de ned on all of That is. the entire boundary of must be glued to X For example if X is a circle then. Figure 1 i shows one possible result of attaching a 1 cell to X Attaching a 1 cell. to X cannot lead to the space illustrated in Figure 1 ii since the entire boundary. of the 1 cell has not been glued to X, We are now ready for our main de nition A nite cell complex is any topo.

logical space X such that there exists a nite nested sequence. 1 X0 X1 Xn X, such that for each i 0 1 2 n Xi is the result of attaching a cell to X i 1. Note that this de nition requires that X0 be a 0 cell If X is a cell complex we. refer to any sequence of spaces as in 1 as a cell decomposition of X Suppose that. Figure 1 On the left a 1 cell is attached to a circle This is not true for the. picture on the right,LECTURE 1 DISCRETE MORSE THEORY 139. X0 X1 X2 X3,Figure 2 A cell decomposition of the torus. in the cell decomposition 1 of the n 1 cells that are attached exactly cd are. d cells Then we say that the cell complex X has a cell decomposition consisting. of cd d cells for every d, We note that a closed d simplex is a d cell Thus a nite simplicial complex. is a cell complex and has a cell decomposition in which the cells are precisely the. closed simplices, In Figure 2 we demonstrate a cell decomposition of a 2 dimensional torus which.

beginning with the 0 cell requires attaching two 1 cells and then one 2 cell Here we. can see one of the most compelling reasons for expanding our view from simplicial. complexes to more general cell complexes Every simplicial decomposition of the. 2 torus has at least 7 vertices 21 edges and 14 triangles. It may seem that quite a bit has been lost in the transition from simplicial. complexes to general cell complexes After all a simplicial complex is completely. described by a nite amount of combinatorial data On the other hand the con. struction of a cell decomposition requires the choice of a number of continuous. maps However if one is only concerned with the homotopy type of the resulting. cell complex then things begin to look a bit more manageable Namely the homo. topy type of X f depends only on the homotopy type of X and the homotopy. class of f, Theorem 1 Let h X X denote a homotopy equivalence a cell and f1. X f2 X two continuous maps If h f1 is homotopic to f2 then. X f1 and X f2 are homotopy equivalent, See Theorem 2 3 on page 120 of 68 An important special case is when h is the. identity map We state this case separately for future reference. Corollary 2 Let X be a topological space a cell and f1 f2 X two. continuous maps If f1 and f2 are homotopic then X f1 and X f2 are. homotopy equivalent, Therefore the homotopy type of a cell complex is determined by the homotopy. classes of the attaching maps Since homotopy clases are discrete objects we have. now recaptured a bit of the combinatorial atmosphere that we seemingly lost when. generalizing from simplicial complexes to cell complexes. Let us now present some examples, 1 Suppose X is a topological space which has a cell decomposition consisting. of exactly one 0 cell and one d cell Then X has a cell decomposition X0. X1 X The space X0 must be the 0 cell and X X1 is the result of attaching. the d cell to X0 Since X0 consists of a single point the only possible attaching. map is the constant map Thus X is constructed from taking a closed d ball and. 140 R FORMAN COMB DIFFERENTIAL TOPOLOGY AND GEOMETRY. identifying all of the points on its boundary One can easily see that this implies. that the resulting space is a d sphere, 2 Suppose X is a topological space which has a cell decomposition consisting of.

exactly one 0 cell and n d cells Then X has a cell decomposition as in 1 such that. X0 is the 0 cell and for each i 1 2 n the space Xi is the result of attaching. a d cell to X i 1 From the previous example we know that X1 is a d sphere. The space X2 is constructed by attaching a d cell to X1 The attaching map is a. continuous map from a d 1 sphere to X1 Every map of the d 1 sphere into. X1 is homotopic to a constant map since d 1 X1 d 1 S d. attaching map is actually a constant map then it is easy to see that the space X2. is the wedge of two d spheres denoted by S d S d The wedge of a collection of. topological spaces is the space resulting from choosing a point in each space taking. the disjoint union of the spaces and identifying all of the chosen points Since the. attaching map must be homotopic to a constant map Corollary 2 implies that X2. is homotopy equivalent to a wedge of two d spheres. When constructing X3 by attaching a d cell to X2 the relevant information is a. map from S d 1 to X2 and the homotopy type of the resulting space is determined. by the homotopy class of this map All such maps are homotopic to a constant. map since d 1 X2 d 1 S d S d,0 Since X2 is homotopy equivalent to. a wedge of two d spheres and the attaching map is homotopic to a constant map. it follows from Theorem 1 that X3 is homotopy equivalent to the space that would. result from attaching a d cell to S d S d via a constant map i e X3 is homotopy. equivalent to a wedge of three d spheres, Continuing in this fashion we can see that X must be homotopy equivalent to. a wedge of n d spheres, The reader should not get the impression that the homotopy type of a cell com. plex is determined by the number of cells of each dimension This is true only for. very few spaces and the reader might enjoy coming up with some other examples. The fact that wedges of spheres can in fact be identi ed by this numerical data. partly explains why the main theorem of many papers in combinatorial topology. is that a certain simplicial complex is homotopy equivalent to a wedge of spheres. Namely such complexes are the easiest to recognize However that does not ex. plain why so many simplicial complexes that arise in combinatorics are homotopy. equivalent to a wedge of spheres I have often wondered if perhaps there is some. deeper explanation for this, 3 Suppose that X is a cell complex which has a cell decomposition consisting. of exactly one 0 cell one 1 cell and one 2 cell Let us consider a cell decomposition. for X with these cells X0 X1 X2 X We know that X0 is the 0 cell. Suppose that X1 is the result of attaching the 1 cell to X0 Then X1 must be a. circle and X2 arises from attaching a 2 cell to X1 The attaching map is a map. from the boundary of the 2 cell i e a circle to X1 which is also a circle Up to. homotopy such a map is determined by its winding number which can be taken. to be a nonnegative integer If the winding number is 0 then wit. Topics in Combinatorial Di erential Topology and Geometry Robin Forman Many questions from a variety of areasof mathematics lead one to the problem of analyzing the topology or the combinatorial geometry of a simplicial complex We will see a number of examples in these notes Some very general theories have

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