Abstract Covering maps and fibrations of spaces fulfilling certain technical conditions are known to satisfy a multiplicative formula relating the Euler characteristic of the do- main to that of the codomain. An open question posed by Albrecht Dold in 1980 asks in general when this is true and can be stated as: for which classes of maps is it true that if χ(X) denotes the Euler characteristic of a space X, and if f : X ! Y has the property that χ(f −1(y)) = k for all y 2 Y and for some integer k, the multiplicative formula χ(X) = k · χ(Y ) holds? A corroborative answer is given herein for simplicial maps of finite simplicial complexes, while counterexamples are constructed for cellular maps of finite CW complexes, continuous maps of closed topological manifolds, and even smooth maps of smooth manifolds. c Copyright by Kelley Brook Johnson 2015 All rights reserved. Acknowledgement I want to express my deepest appreciation to everyone who has helped me with this thesis and supported me throughout my years at Tulane. Thank you, Slawomir Kwasik, for your patience, support, and wonderful advice on matters of acedemia, love, and life. It has been a pleasure working with you. Thank you to everyone in the mathematics department at Tulane for all of your help and for making our department a second home to me. Special thanks to my committee members Albert Vitter, Maurice Dupre, Michael Mislove, and Rafal Komendarczyk. Also to Andreas Michaelides, Ellis Fenske, and Franz Hoffman, not only for their help on this project, but for their friendship during our time in New Orleans. Lastly, thank you to Casey Tatangelo, my parents Douglas and Dana Johnson, and my family and friends for your love and support. ii List of Figures 2.1 Homotopy Lifting Property Diagram [1].................7 4.1 Graph of g : I ! I ............................ 17 6.1 Graph of f : R ! R ............................ 29 6.2 Graph of α, β, and f ........................... 30 iii Contents Acknowledgement ii List of Figures iii 1 Introduction1 2 Established Conditions for Multiplicativity5 3 Simplicial Maps 11 4 Cellular Maps 16 5 A Map of Topological Manifolds 24 6 Smooth Maps 26 References 35 iv 1 Chapter 1 Introduction In 1758 Euler published a simple yet remarkable formula which states that for a convex polyhedron: V − E + F = 2; (1.1) where V , E, and F are the number of polyhedral vertices, edges, and faces, respec- tively. It was very surprising at the time that this very basic formula should apply to all convex polyhedra and that any triangulation may be used to find V, E, and F. Today, many topology students learn this formula as a special case of the Euler characteristic. The modern definition, given below extends the notion of the polyhedral formula to a much larger class of spaces, topological spaces of any dimension that are of bounded finite type (meaning a space whose homology groups are all finitely generated, with only finitely many being nonzero). Definition 1.1. The Euler characteristic of a space of bounded finite type, X, is 1 P i th χ(X) = (−1) Rank(Hi(X)), where Hi(X) is the i homology group of X. i=1 It is evident from this definition that the Euler characteristic depends only on the 2 homology of X. Spaces that are homeomorphic or homotopy equivalent have isomor- phic homology groups, so homeomorphic spaces and homotopy equivalent spaces have the same Euler characteristic. Thus it is a topological and homotopical invariant [2]. Being able to encode topological data in a single integer via the Euler character- istic has proven very useful. There have been applications of Euler characteristic, not only in pure mathematics, such as in the proof of the Lefschetz fixed point theorem, or in classifying surfaces, but also in applied math and science. For example, as stated in [3], the Euler characteristic has been a valuable tool for target ennumeration in sensor networks, and an \important parameter in the physics of porous materials." Fortunately, the Euler characteristic has many properties that make its computation for complicated spaces easier by using the Euler characteristics of simpler related spaces. For example, if the topological space X is the union of subspaces A and B, then χ(X) = χ(A) + χ(B) − χ(A \ B), provided that X, A, and B are of bounded finite type. Additionally, χ(A × B) = χ(A) × χ(B)[2]. If instead, X is the connected sum of closed manifolds A and B of dimension n, then χ(X) = χ(A) + χ(B) − χ(Sn). In 1980, Albrecht Dold posed the following question about the multiplicativity of the Euler characteristic, a question which has remained open for almost 35 years [4]. Question 1.1. Motivated by the Vietoris mapping theorem and analogous results, we ask whether the following is true: \if f : X ! Y is a continuous map between compact metric spaces such that χ(f −1(y)) = 1 for all y 2 Y then χ(X) = χ(Y )." The question is of interest also for simpler spaces X; Y , say compact CW-spaces or manifolds. On the other hand, the question loses much of its interest if the continuous map f is further restricted (for example simplicial or fibration). Answers for restricted 3 classes of maps will not be counted towards a solution! A little more general, one can ask whether χ(X) = k · χ(Y ) (1.2) if χ(f −1(y)) has the same value k 2 Z for all y 2 Y . Here is a brief description of our results. In chapter 2 we recall well known conditions which imply a positive answer to Dold's question. These include coverings (finite) and fibrations. In chapter 3 we provide the following new result for simplicial maps of finite sim- plicial complexes: Theorem 1.2. Let f : X ! Y be a simplicial map of finite simplicial complexes such that χ(f −1(y)) = k 8y 2 Y . Then χ(X) = k · χ(Y ): (1.3) This gives a positive answer to Dold's question for a class of maps which, despite being restricted, allows more variation among preimages and is closely related to cellular and smooth classes of maps. The argument depends heavily on the special properties of simplicial maps. Chapter 4 gives a negative answer to Dold's question even for cellular maps of CW- complexes. A counterexample to the proposed multiplicative formula is constructed 4 for each integer k, where k is the Euler characteristic of the preimage of any point under the specified cellular map. The results are then summarized by the following theorem: Theorem 1.3. For any k 2 Z, there is a cellular map f : X ! Y of finite CW complexes X; Y such that χ(f −1(y)) = k 8y 2 Y and χ(X) 6= k · χ(Y ). This is in sharp contrast with Theorem 1.2. In Chapter 5 we present an example of a map of closed topological manifolds un- der which the preimage of each point has identical Euler characteristic equal to 3, but does not satisfy equation (1.2), thus giving a counterexample to the formula in Dold's question for maps between members of a very nice class of spaces. Lastly, we refine one of our counterexamples for CW-complexes to obtain a smooth map between smooth manifolds, hence obtaining the following theorem, which is un- expected in view of the result for simplicial maps: Theorem 1.4. There is a smooth map f : X ! Y of smooth manifolds X; Y such that χ(f −1(y)) = k 8y 2 Y and χ(X) 6= k · χ(Y ). 5 Chapter 2 Established Conditions for Multiplicativity An alternate formulation of the Euler characteristic definition is given for CW com- plexes by the subsequent theorem due to Euler and Poincare [1]: Theorem 2.1. For a finite CW complex, X 1 X i χ(X) = (−1) ci (2.1) i=1 where ci is the number of i-cells in X. This characterization is often more convenient, since, at times, counting numbers of cells can be considerably easier than computing homology. Theorem 2.3 is a nice result about the multiplicativity of the Euler characteristic for covering maps that most first year algebraic topology students are familiar with and whose proof (as outlined in [1]) makes use of this new Euler characteristic formula. First, recall the definition of a covering map: Definition 2.2. Let X and Y be topological spaces. A map p : X ! Y is a 6 covering map if for each point y 2 Y there is an open neighborhood U of y in Y such that p−1(U) is a union of disjoint open sets in X, each of which is mapped homeomorphically onto U by p. If there are k such disjoint open sets in X, then X is called a k-sheeted covering of Y . So a k-sheeted covering of a finite CW complex Y , can be thought of locally as a disjoint union of k homeomorphic copies of a neighborhood of any point in Y . Theorem 2.3. If X is a k-sheeted covering of a finite CW complex, Y , then χ(X) = k · χ(Y ) Proof. It follows from the Lifting Lemma [5] that for each i-cell in Y , the characteristic i map from the unit disc into Y , φα : D ! Y , lifts to a map whose image is contained in a specified homeomorphic copy of that cell, contained in X. Since there are k copies to choose from, the characteristic map from the i-dimensional unit disc, Di, to Y lifts exactly k ways into X.