Advanced Fixed Point Theory for Economics. Andrew McLennan


 Archibald Miles
 2 years ago
 Views:
Transcription
1 Advanced Fixed Point Theory for Economics Andrew McLennan April 8, 2014
2 Preface Over two decades ago now I wrote a rather long survey of the mathematical theory of fixed points entitled Selected Topics in the Theory of Fixed Points. It had no content that could not be found elsewhere in the mathematical literature, but nonetheless some economists found it useful. Almost as long ago, I began work on the project of turning it into a proper book, and finally that project is coming to fruition. Various events over the years have reinforced my belief that the mathematics presented here will continue to influence the development of theoretical economics, and have intensified my regret about not having completed it sooner. There is a vast literature on this topic, which has influenced me in many ways, and which cannot be described in any useful way here. Even so, I should say something about how the present work stands in relation to three other books on fixed points. Fixed Point Theorems with Applications to Economics and Game Theory by Kim Border (1985) is a complement, not a substitute, explaining various forms of the fixed point principle such as the KKMS theorem and some of the many theorems of Ky Fan, along with the concrete details of how they are actually applied in economic theory. Fixed Point Theory by Dugundji and Granas (2003) is, even more than this book, a comprehensive treatment of the topic. Its fundamental point of view (applications to nonlinear functional analysis) audience (professional mathematicians) and technical base (there is extensive use of algebraic topology) arequitedifferent, butitisstillaworkwithmuchtooffertoeconomics. Particularly notable is the extensive and meticulous information concerning the literature and history of the subject, which is full of affection for the theory and its creators. The book that was, by far, the most useful to me, is The Lefschetz Fixed Point Theorem by Robert Brown (1971). Again, his approach and mine have differences rooted in the nature of our audiences, and the overall objectives, but at their cores the two books are quite similar, in large part because I borrowed a great deal. I would like to thank the many people who, over the years, have commented favorably on Selected Topics. It is a particular pleasure to acknowledge some very detailed and generous written comments by Klaus Ritzberger. This work would not have been possible without the support and affection of my families, both present and past, for which I am forever grateful. i
3 Contents 1 Introduction and Summary The First Fixed Point Theorems Fixing Kakutani s Theorem Essential Sets of Fixed Points Index and Degree Manifolds The Degree The Fixed Point Index Topological Consequences Dynamical Systems I Topological Methods 22 2 Planes, Polyhedra, and Polytopes Affine Subspaces Convex Sets and Cones Polyhedra Polytopes Polyhedral Complexes Graphs Computing Fixed Points The LemkeHowson Algorithm Implementation and Degeneracy Resolution Using Games to Find Fixed Points Sperner s Lemma The Scarf Algorithm Homotopy Remarks on Computation Topologies on Spaces of Sets Topological Terminology Spaces of Closed and Compact Sets Vietoris Theorem Hausdorff Distance Basic Operations on Subsets ii
4 CONTENTS iii Continuity of Union Continuity of Intersection Singletons Continuity of the Cartesian Product The Action of a Function The Union of the Elements Topologies on Functions and Correspondences Upper and Lower Semicontinuity The Strong Upper Topology The Weak Upper Topology The Homotopy Principle Continuous Functions Metric Space Theory Paracompactness Partitions of Unity Topological Vector Spaces Banach and Hilbert Spaces EmbeddingTheorems Dugundji s Theorem Retracts Kinoshita s Example Retracts Euclidean Neighborhood Retracts Absolute Neighborhood Retracts Absolute Retracts Domination Essential Sets of Fixed Points The FanGlicksberg Theorem Convex Valued Correspondences Kinoshita s Theorem Approximation of Correspondences The Approximation Result Extending from the Boundary of a Simplex Extending to All of a Simplicial Complex Completing the Argument II Smooth Methods Differentiable Manifolds Review of Multivariate Calculus Smooth Partitions of Unity
5 CONTENTS Manifolds Smooth Maps Tangent Vectors and Derivatives Submanifolds Tubular Neighborhoods Manifolds with Boundary Classification of Compact 1Manifolds Sard s Theorem Sets of Measure Zero A Weak Fubini Theorem Sard s Theorem Measure Zero Subsets of Manifolds Genericity of Transversality Degree Theory Orientation Induced Orientation The Degree Composition and Cartesian Product The Fixed Point Index Axioms for an Index on a Single Space Multiple Spaces The Index for Euclidean Spaces Extension by Commutativity Extension by Continuity III Applications and Extensions Topological Consequences Euler, Lefschetz, and EilenbergMontgomery The Hopf Theorem More on Maps Between Spheres Invariance of Domain Essential Sets Revisited Vector Fields and their Equilibria Euclidean Dynamical Systems Dynamics on a Manifold The Vector Field Index Dynamic Stability The Converse Lyapunov Problem A Necessary Condition for Stability
6 Chapter 1 Introduction and Summary The Brouwer fixed point theorem states that if C is a nonempty compact convex subset of a Euclidean space and f : C C is continuous, then f has a fixed point, which is to say that there is an x C such that f(x ) = x. The proof of this by Brouwer (1912) was one of the major events in the history of topology. Since then the study of such results, and the methods used to prove them, has flourished, undergoing radical transformations, becoming increasingly general and sophisticated, and extending its influence to diverse areas of mathematics. Around 1950, most notably through the work of Nash (1950, 1951) on noncooperative games, and the work of Arrow and Debreu (1954) on general equilibrium theory, it emerged that in economists most basic and general models, equilibria are fixed points. The most obvious consequence of this is that fixed point theorems provide proofs that these models are not vacuous. But fixed point theory also informs our understanding of many other issues such as comparative statics, robustness under perturbations, stability of equilibria with respect to dynamic adjustment processes, and the algorithmics and complexity of equilibrium computation. In particular, since the mid 1970 s the theory of games has been strongly influenced by refinement concepts defined largely in terms of robustness with respect to certain types of perturbations. As the range and sophistication of economic modelling has increased, more advanced mathematical tools have become relevant. Unfortunately, the mathematical literature on fixed points is largely inaccessible to economists, because it relies heavily on homology. This subject is part of the standard graduate school curriculum for mathematicians, but for outsiders it is difficult to penetrate, due to its abstract nature and the amount of material that must be absorbed at the beginning before the structure, nature, and goals of the theory begin to come into view. Many researchers in economics learn advanced topics in mathematics as a side product of their research, but unlike infinite dimensional analysis or continuous time stochastic processes, algebraic topology will not gradually achieve popularity among economic theorists through slow diffusion. Consequently economists have been, in effect, shielded from some of the mathematics that is most relevant to their discipline. This monograph presents an exposition of advanced material from the theory of fixed points that is, in several ways, suitable for graduate students and researchers in mathematical economics and related fields. In part the fit with the intended 2
7 1.1. THE FIRST FIXED POINT THEOREMS 3 audience is a matter of coverage. Economic models always involve domains that are convex, or at least contractible, so there is little coverage here of topics that only become interesting when the underlying space is more complicated. For the settings of interest, the treatment is comprehensive and maximally general, with issues related to correspondences always in the foreground. The project was originally motivated by a desire to understand the existence proofs in the literature on refinements of Nash equilibrium as applications of preexisting mathematics, and the continuing influence of this will be evident. The mathematical prerequisites are within the common background of advanced students and researchers in theoretical economics. Specifically, in addition to multivariate calculus and linear algebra, we assume that the reader is familiar with basic aspects of pointset topology. What we need from topics that may be less familiar to some (e.g., simplicial complexes, infinite dimensional linear spaces, the theory of retracts) will be explained in a selfcontained manner. There will be no use of homological methods. The avoidance of homology is a practical necessity, but it can also be seen as a feature rather than a bug. In general, mathematical understanding is enhanced when brute calculations are replaced by logical reasoning based on conceptually meaningful definitions. To say that homology is a calculational machine is a bit simplistic, but it does have that potential in certain contexts. Avoiding it commits us to work with notions that have more direct and intuitive geometric content. (Admittedly there is a slight loss of generality, because there are acyclic that is, homologically trivial spaces that are not contractible, but this is unimportant because such spaces are not found in nature. ) Thus our treatment of fixed point theory can be seen as a mature exposition that presents the theory in a natural and logical manner. In the remainder of this chapter we give a broad overview of the contents of the book. Unlike many subjects in mathematics, it is possible to understand the statements of many of the main results with much less preparation than is required to understand the proofs. Needless to say, as usual, not bothering to study the proofs has many dangers. In addition, the material in this book is, of course, closely related to various topics in theoretical economics, and in many ways quite useful preparation for further study and research. 1.1 The First Fixed Point Theorems A fixed point of a function f : X X is an element x X such that f(x ) = x. If X is a topological space, it is said to have the fixed point property if every continuous function from X to itself has a fixed point. The first and most famous result in our subject is Brouwer s fixed point theorem: Theorem (Brouwer (1912)). If C R m is nonempty, compact, and convex, then it has the fixed point property. Chapter 3 presents various proofs of this result. Although some are fairly brief, none of them can be described as truly elementary. In general, proofs of Brouwer s
8 4 CHAPTER 1. INTRODUCTION AND SUMMARY theorem are closely related to algorithmic procedures for finding approximate fixed points. Chapter 3 discusses the best known general algorithm due to Scarf, a new algorithm due to the author and Rabee Tourky, and homotopy methods, which are the most popular in practice, but require differentiability. The last decade has seen major breakthroughs in computer science concerning the computational complexity of computing fixed points, with particular reference to (seemingly) simple games and general equilibrium models. These developments are sketched briefly in Section 3.7. In economics and game theory fixed point theorems are most commonly used to prove that a model has at least one equilibrium, where an equilibrium is a vector of endogenous variable for the model with the property that each individual agent s predicted behavior is rational, or utility maximizing, if that agent regards all the other endogenous variables as fixed. In economics it is natural, and in game theory unavoidable, to consider models in which an agent might have more than one rational choice. Our first generalization of Brouwer s theorem addresses this concern. If X and Y are sets, a correspondence F : X Y is a function from X to the nonempty subsets of Y. (Ontherareoccasions when theyarise, we use theterm set valued mapping forafunction fromx toall thesubsets ofy, including theempty set.) We will tend to regard a function as a special type of correspondence, both intuitively and in the technical sense that we will frequently blur the distinction between a function f : X Y and the associated correspondence x {f(x)}. If Y is a topological space, F is compact valued if, for all x X, F(x) is compact. Similarly, if Y is a subset of a vector space, then F is convex valued if each F(x) is convex. The extension of Brouwer s theorem to correspondences requires a notion of continuity for correspondences. If X and Y are topological spaces, a correspondence F : X Y is upper semicontinuous if it is compact valued and, for each x 0 X and each neighborhood V Y of F(x 0 ), there is a neighborhood U X of x 0 such that F(x) V for all x U. It turns out that if X and Y are metric spaces and Y is compact, then F is upper semicontinuous if and only if its graph Gr(F) := {(x,y) X Y : y F(x)} is closed. (Proving this is a suitable exercise, if you are so inclined.) Thinking of upper semicontinuity as a matter of the graph being closed is quite natural, and in economics this condition is commonly taken as definition, as in Debreu (1959). In Chapter 5 we will develop a topology on the space of nonempty compact subsets of Y such that F is upper semicontinuous if and only if it is a continuous function relative to this topology. A fixed point of a correspondence F : X X is a point x X such that x F(x ). Kakutani (1941) was motivated to prove the following theorem by the desire to provide a simple approach to the von Neumann (1928) minimax theorem, which is a fundamental result of game theory. This is the fixed point theorem that is most commonly applied in economic analysis.
9 1.2. FIXING KAKUTANI S THEOREM 5 Theorem (Kakutani s Fixed Point Theorem). If C R m is nonempty, compact, and convex, and F : C C is an upper semicontinuous convex valued correspondence, then F has a fixed point. 1.2 Fixing Kakutani s Theorem Mathematicians strive to craft theorems that maximize the strength of the conclusions while minimizing the strength of the assumptions. One reason for this is obvious: a stronger theorem is a more useful theorem. More important, however, is the desire to attain a proper understanding of the principle the theorem expresses, and to achieve an expression of this principle that is unencumbered by useless clutter. When a theorem that is too weak is proved using methods that happen to work there is a strong suspicion that attempts to improve the theorem will uncover important new concepts. In the case of Brouwer s theorem the conclusion, that the space has the fixed point property, is a purely topological assertion. The assumption that the space is convex, and in Kakutani s theorem the assumption that the correspondence s values are convex, are geometric conditions that seems out of character and altogether too strong. Suitable generalizations were developed after World War II. A homotopy is a continuous function h : X [0,1] Y where X and Y are topological spaces. It is psychologically natural to think of the second variable in the domain as representing time, and we let h t := h(,t) : X Y denote the function at time t, so that h is a process that continuously deforms a function h 0 into h 1. Another intuitive picture is that h is a continuous path in the space C(X,Y) of continuous function from X to Y. As we will see in Chapter 5, this intuition can be made completely precise: when X and Y are metric spaces and X is compact, there is a topology on C(X,Y) such that a continuous path h : [0,1] C(X,Y) is the same thing as a homotopy. We say that two functions f,g : X Y are homotopic if there is a homotopy h with h 0 = f and h 1 = g. This is easily seen to be an equivalence relation: symmetry and reflexivity are obvious, and to establish transitivity we observe that if e is homotopic to f and f is homotopic to g, then there is a homotopy between e and g that follows a homotopy between e and f at twice its original speed, then follows a homotopy between f and g at double the pace. The equivalence classes are called homotopy classes. AspaceX iscontractibleiftheidentityfunctionid X ishomotopictoaconstant function. That is, there is a homotopy c : X [0,1] X such that c 0 = Id X and c 1 (X) is a singleton; such a homotopy is called a contraction. Convex sets are contractible. More generally, a subset X of a vector space is starshaped if there is x X (the star) such that X contains the line segment {(1 t)x+tx : 0 t 1} between each x X and x. If X is starshaped, there is a contraction (x,t) (1 t)x+tx.
10 6 CHAPTER 1. INTRODUCTION AND SUMMARY It seems natural to guess that a nonempty compact contractible space has the fixedpointproperty. Whether thisisthecasewasanopenproblemforseveral years, but it turns out to be false. In Chapter 7 we will see an example due to Kinoshita (1953) of a nonempty compact contractible subset of R 3 that does not have the fixed point property. Fixed point theory requires some additional ingredient. If X is a topological space, a subset A X is a retract if there is a continuous function r : X A with r(a) = a for all a A. Here we tend to think of X as a simple space, and the hope is that although A might seem to be more complex, or perhaps crumpled up, it nonetheless inherits enough of the simplicity of X. A particularly important manifestation of this is that if r : X A is a retraction and X has the fixed point property, then so does A, because if f : A A is continuous, then so is f r : X A X, so f r has a fixed point, and this fixed point necessarily lies in A and is consequently a fixed point of f. Also, a retract of a contractible space is contractible because if c : X [0,1] X is a contraction of X and r : X A X is a retraction, then (a,t) r(c(a,t)) is a contraction of A. A set A R m is a Euclidean neighborhood retract (ENR) if there is an open superset U R m of A and a retraction r : U A. If X and Y are metric spaces, anembedding of X in Y is a functione : X Y that is a homeomorphism between X and e(x). That is, e is a continuous injection 1 whose inverse is also continuous when e(x) has the subspace topology inherited from Y. An absolute neighborhood retract (ANR) is a separable 2 metric space X such that whenever Y is a separable metric space and e : X Y is an embedding, there is an open superset U Y of e(x) and a retraction r : U e(x). This definition probably seems completely unexpected, and it s difficult to get any feeling for it right away. In Chapter 7 we ll see that ANR s have a simple characterization, and that many of the types of spaces that come up most naturally are ANR s, so this condition is quite a bit less demanding than one might guess at first sight. In particular, it will turn out that every ENR is an ANR, so that being an ENR is an intrinsic property insofar as it depends on the topology of the space and not on how the space is embedded in a Euclidean space. An absolute retract (AR) is a separable metric space X such that whenever Y is a separable metric space and e : X Y is an embedding, there is a retraction r : Y e(x). In Chapter 7 we will prove that an ANR is an AR if and only if it is contractible. Theorem If C is a nonempty compact AR and F : C C is an upper semicontinuous contractible valued correspondence, then F has a fixed point. An important point is that the values of F are not required to be ANR s. 1 We will usually use the terms injective rather than onetoone, surjective rather than onto, and bijective to indicate that a function is both injective and surjective. An injection is an injective function, a surjection is a surjective function, and a bijection is a bijective function. 2 A metric space is separable if it has a countable dense subset.
11 1.3. ESSENTIAL SETS OF FIXED POINTS 7 For practical purposes this is the maximally general topological fixed point theorem, but for mathematicians there is an additional refinement. There is a concept called acyclicity that is defined in terms of the concepts of algebraic topology. A contractible set is necessarily acyclic, but there are acyclic spaces (including compact ones) that are not contractible. The famous EilenbergMontgomery fixed point theorem is: Theorem (Eilenberg and Montgomery (1946)). If C is a nonempty compact AR and F : C C is an upper semicontinuous acyclic valued correspondence, then F has a fixed point. 1.3 Essential Sets of Fixed Points It might seem like we have already reached a satisfactory and fitting resolution of The Fixed Point Problem, but actually (both in pure mathematics and in economics) this is just the beginning. You see, fixed points come in different flavors s t 1 Figure 1.1 The figure above shows a function f : [0,1] [0,1] with two fixed points, s and t. If we perturb the function slightly by adding a small positive constant, s disappears in the sense that the perturbed function does not have a fixed point anywhere near s, but a function close to f has a fixed point near t. More precisely, if X is a topological space and f : X X is continuous, a fixed point x of f is essential if, for any neighborhood U of x, there is a neighborhood V of the graph of f such that any continuous f : X X whose graph is contained in V has a fixed point in U. If a fixed point is not essential, then we say that it is inessential. These concepts were introduced by Fort (1950). There need not be an essential fixed point. The function shown in Figure 1.2
12 8 CHAPTER 1. INTRODUCTION AND SUMMARY has an interval of fixed points. If we shift the function down, there will be a fixed point near the lower endpoint of this interval, and if we shift the function up there will be a fixed point near the upper endpoint. This example suggests that we might do better to work with sets of fixed points. A set S of fixed points of a function f : X X is essential if it is closed, it has a neighborhood that contains no other fixed points, and for any neighborhood U of S, there is a neighborhood V of the graph of f such that any continuous f : X X whose graph is contained in V has a fixed point in U. The problem with this concept is that large connected sets are not of much use. For example, if X is compact and has the fixed point property, then the set of all fixed points of f is essential. It seems that we should really be interested in sets of fixed points that are either essential and connected 3 or essential and minimal in the sense of not having a proper subset that is also essential Figure 1.2 In Chapter 8 we will show that any essential set of fixed points contains a minimal essential set, and that minimal essential sets are connected. The theory of refinements of Nash equilibrium (e.g., Selten (1975); Myerson (1978); Kreps and Wilson (1982); Kohlberg and Mertens (1986); Mertens (1989, 1991); Govindan and Wilson (2008)) has many concepts that amount to a weakening of the notion of essential set, insofar as the set is required to be robust with respect to only certain types of perturbations of the function or correspondence. In particular, Jiang (1963) pioneered the application of the concept to game theory, defining an essential!nash equilibrium and an essential set of Nash equilibria in terms of robustness with respect to perturbations of the best response correspondence induced by perturbations of the payoffs. The mathematical foundations of such 3 We recall that a subset S of a topological space X is connected if there do not exist two disjoint open sets U 1 and U 2 with S U 1 S U 2 and S U 1 U 2.
13 1.4. INDEX AND DEGREE 9 concepts are treated in Section Index and Degree There are different types of essential fixed points. Figure 1.3 shows a function with three fixed points. At two of them the function starts above the diagonal and goes below it as one goes from left to right, and at the third it is the other way around. For any k it is easy to imagine a function with k fixed points of the first type and k 1 fixed points of the second type. This phenomenon generalizes to higher dimensions. Let D m = {x R m : x 1} and S m 1 = {x R m : x = 1} be the mdimensional unit disk and the (m 1)dimensional unit sphere, and suppose that f : D m D m is a C function. In the best behaved case each fixed point x is in the interior D m \ S m 1 of the disk and regular, which means that Id R m Df(x ) is nonsingular, where Df(x ) : R m R m is the derivative of f at x. We define the index of x to be 1 if the determinant of Id R m Df(x ) is positive and 1 if this determinant is negative. We will see that there is always one more fixed point of index 1 than there are fixed points of index 1, which is to say that the sum of the indices is 1. What about fixed points on the boundary of the disk, or fixed points that aren t regular, or nontrivial connected sets of fixed points? What about correspondences? What happens if the domain is a possibly infinite dimensional ANR? The most challenging and significant aspect of our work will be the development of an axiomatic theory of the index that is general enough to encompass all these possibilities. The work proceeds through several stages, and we describe them in some detail now Figure 1.3
14 10 CHAPTER 1. INTRODUCTION AND SUMMARY Manifolds First of all, it makes sense to expand our perspective a bit. An mdimensional manifold is a topological space that resembles R m in a neighborhood of each of its points. More precisely, for each p M there is an open U R m and an embedding ϕ : U M whose image is open and contains p. Such a ϕ is a parameterization and its inverse is a coordinate chart. The most obvious examples are R m itself and S m. If, in addition, N is an ndimensional manifold, then M N is an (m+n) dimensional manifold. Thus the torus S 1 S 1 is a manifold, and this is just the most easily visualized member of a large class of examples. An open subset of an mdimensional manifold is an mdimensional manifold. A 0dimensional manifold is just a set with the discrete topology. The empty set is a manifold of any dimension, including negative dimensions. Of course these special cases are trivial, but they come up in important contexts. A collection {ϕ i : U i M} i I of parameterizations is an atlas if its images cover M. The composition ϕ 1 j ϕ i (with the obvious domain of definition) is called a transition function. If, for some 1 r, all the transition functions are C r functions, then the atlas is a C r atlas. An mdimensional C r manifold is an mdimensional manifold together with a C r atlas. The basic concepts of differential and integral calculus extend to this setting, leading to a vast range of mathematics. In our formalities we will always assume that M is a subset of a Euclidean space R k called the ambient space, and that the parameterizations ϕ i and the coordinate charts ϕ 1 i are C r functions. This is a bit unprincipled for example, physicists see only the universe, and their discourse is more disciplined if it does not refer to some hypothetical ambient space but this maneuver is justified by embedding theorems due to Whitney that show that it does not entail any serious loss of generality. The advantages for us are that this approach bypasses certain technical pathologies while allowing for simplified definitions, and in many settings the ambient space will prove quite handy. For example, a function f : M N (where N is now contained in some R l ) is C r for our purposes if it is C r in the standard sense: for any S R k a function h : S R l is C r, by definition, if there is an open W R k containing S and a C r function H : W R l such that h = H S. Having an ambient space around makes it relatively easy to establish the basic objects and facts of differential calculus. Suppose that ϕ i : U i M is a C r parameterization. If x U i and ϕ i (x) = p, the tangent space of M at p, which we denote by T p M, is the image of Dϕ i (x). This is an mdimensional linear subspace of R k. If f : M N is C r, the derivative Df(p) : T p M T f(p) N of f at p is the restriction to T p M of the derivative DF(p) of any C r function F : W R l defined on an open W R k containing M whose restriction to M is f. (In Chapter 10 we will show that the choice of F doesn t matter.) The chain rule holds: if, in addition, P is a pdimensional C r manifold and g : N P is a C r function, then g f is C r and D(g f)(p) = Dg(f(p)) Df(p) : T p M T g(f(p)) P.
15 1.4. INDEX AND DEGREE 11 The inverse and implicit function theorems have important generalizations. The point p is a regular point of f if the image of Df(p) is all of T f(p) N. We say that f : M N is a C r diffeomorphism if m = n, f is a bijection, and both f and f 1 are C r. The generalized inverse function theorem asserts that if m = n, f : M N is C r, and p is a regular point of f, then there is an open U M containing p such that f(u) is an open subset of N and f U : U f(u) is a C r diffeomorphism. If 0 s m, a set S R k is an sdimensional C r submanifold of M if it is an sdimensional C r submanifold that happens to be contained in M. We say that q N is a regular value of f if every p f 1 (q) is a regular point. The generalized implicit function theorem, which is known as the regular value theorem, asserts that if q is a regular value of f, then f 1 (q) is an (m n)dimensional C r submanifold of M The Degree The degree is closely related to the fixed point index, but it has its own theory, which has independent interest and significance. The approach we take here is to work with the degree up to the point where its theory is more or less complete, then translate what we have learned into the language of the fixed point index. We now need to introduce the concept of orientation. Two ordered bases v 1,...,v m and w 1,...,w m of an mdimensional vector space have the same orientation if the determinant of the linear transformation taking each v i to w i is positive. It is easy to see that this is an equivalence relation with two equivalence classes. An oriented vector space is a finite dimensional vector space with a designated orientation whose elements are said to be positively oriented. If V and W are mdimensional oriented vector spaces, a nonsingular linear transformation L : V W is orientation preserving if it maps positively oriented ordered bases of V to positively oriented ordered bases of W, and otherwise it is orientation reversing. For an intuitive appreciation of this concept just look in a mirror: the linear map taking each point in the actual world to its position as seen in the mirror is orientation reversing, with right shoes turning into left shoes and such. In our discussion of degree theory nothing is lost by working with C objects rather than C r objects for general r, and smooth will be a synonym for C. An orientation for a smooth manifold M is a continuous specification of an orientation of each of the tangent spaces T p M. We say that M is orientable if it has an orientation; the most famous examples of unorientable manifolds are the Möbius strip and the Klein bottle. (From a mathematical point of view 2dimensional projective space is perhaps more fundamental, but it is difficult to visualize.) An oriented manifold is a manifold together with a designated orientation. If M and N are oriented smooth manifolds of the same dimension, f : M N is a smooth map, and p is a regular point of f, we say that f is orientation preserving at p if Df(p) : T p M T f(p) N is orientation preserving, and otherwise f is orientation reversing at p. If q is a regular value of f and f 1 (q) is finite, then the degree of f over q, denoted by deg q (f), is the number of points in f 1 (q) at which f is orientation preserving minus the number of points in f 1 (q) at which f is orientation reversing.
16 12 CHAPTER 1. INTRODUCTION AND SUMMARY We need to extend the degree to situations in which the target point q is not a regular value of f, and to functions that are merely continuous. Instead of being abletodefinethedegreedirectly, aswedidabove, we willneedtoproceedindirectly, showing that the generalized degree is determined by certain of its properties, which we treat as axioms. The first step is to extend the concept, giving it a local character. For a compact C M let C = C (M \C) be the topological boundary of C, and let intc = C\ C be its interior. A smooth function f : C N with compact domain C M is said to be smoothly degree admissible over q N if f 1 (q) C = and q is a regular value of f. As above, for such a pair (f,q) we define deg q (f) to be the number of p f 1 (q) at which f is orientation preserving minus the number of p f 1 (q) at which f is orientation reversing. Note that deg q (f) = deg q (f C ) whenever C is a compact subset of C and f 1 (q) has an empty intersection with the closure of C \ C. Also, if C = C 1 C 2 where C 1 and C 2 are compact and disjoint, then deg q (f) = deg q (f C 1 )+deg q (f C 2 ). From the point of view of topology, what makes the degree important is its invariance under homotopy. If C M is compact, a smooth homotopy h : C [0,1] N is smoothly degree admissible over q if h 1 (q) ( C [0,1]) = and q is a regular value of h 0 and h 1. In this circumstance deg q (h 0) = deg q (h 1). ( ) Figure 1.4 illustrates the intuitive character of the proof t = 0 t = 1 Figure 1.4
17 1.4. INDEX AND DEGREE 13 The notion of an mdimensional manifold with boundary is a generalization of the manifold concept in which each point in the space has a neighborhood that is homeomorphic to an open subset of the closed half space {x R m : x 1 0}. Aside from the half space itself, the closed disk D m = {x R m : x 1} is perhaps the most obvious example, but for us the most important example is M [0,1] where M is an (m 1)dimensional manifold without boundary. Note that any m dimensional manifold without boundary is (automatically and trivially) a manifold with boundary. All elements of our discussion of manifolds generalize to this setting. In particular, the generalization of the regular value theorem states that if M is an mdimensional smooth manifold with boundary, N is an ndimensional (boundaryless) manifold, f : M N is smooth, and q N is a regular value of both f and the restriction of f to the boundary of M, then f 1 (q) is an (m n)dimensional manifold with boundary, its boundary is its intersection with the boundary of M, and at each point in this intersection the tangent space of f 1 (q) is not contained in the tangent space of the boundary of M. In particular, if the dimension of M is the dimension of N plus one, then f 1 (q) is a 1dimensional manifold with boundary. If, in addition, f 1 (q) is compact, then it has finitely many connected components. Suppose now that h : C [0,1] N is smoothly degree admissible over q, and that q is a regular value of h. The consequences of applying the regular value theorem to the restriction of h to intc [0,1] are as shown in Figure 1.4: h 1 (q) is a 1dimensional manifold with boundary, its boundary is its intersection with C {0,1}, and h 1 (q) is not tangent to C {0,1} at any point in this intersection. In addition h 1 (q) is compact, so it has finitely many connected components, each of which is compact. A connected compact 1dimensional manifold with boundary is either a circle or a line segment. (It will turn out that this obvious fact is surprisingly difficult to prove!) Thus each component of h 1 (q) is either a circle or a line segment connecting two points in its boundary. If a line segment connects two points in C {0}, say (p,0) and (p,0), then it turns out that h 0 is orientation preserving at p if and only if it is orientation reversing at p. Similarly, if a line segment connects two points (p,1) and (p,1) in C {1}, then h 1 is orientation preserving at p if and only if it is orientation reversing at p. On the other hand, if a line segment connects a point (p 0,0) in C {0} to a point (p 1,1) in C {1}, then h 0 is orientation preserving at p 0 if and only if h 1 is orientation preserving at p 1. Equation ( ) is obtained by summing these facts over the various components of h 1 (q). This completes our discussion of the proof of ( ) except for one detail: if h : C [0,1] N is a smooth homotopy that is smoothly degree admissible over q, q is not necessarily a regular value of h. Nevertheless, Sard s theorem (which is the subject of Chapter 11, and a crucial ingredient of our entire approach) implies that h has regular values in any neighborhood of q, and it is also the case that deg q (h 0) = deg q (h 0) and deg q (h 1) = deg q (h 1) when q is sufficiently close to q. It turns out that the smooth degree is completely characterized by the properties we have seen. That is, if D (M,N) is the set of pairs (f,q) in which f : C N is smoothly degree admissible over q, then (f,q) deg q (f) is the unique function from D (M,N) to Z satisfying: ( 1) deg q (f) = 1 for all (f,q) D (M,N) such that f 1 (q) is a singleton {p}
18 14 CHAPTER 1. INTRODUCTION AND SUMMARY and f is orientation preserving at p. ( 2) deg q (f) = r i=1 deg q (f Ci ) whenever (f,q) D (M,N), the domain of f is C, and C 1,...,C r are pairwise disjoint compact subsets of C such that f 1 (q) intc 1... intc r. ( 3) deg q (h 0) = deg q (h 1) whenever C M is compact and the homotopy h : C [0,1] N is smoothly degree admissible over q. We note two additional properties of the smooth degree. The first is that if, in addition to M and N, M and N are m dimensional smooth functions, (f,q) D (M,N), and (f,q ) D (M,N ), then (f f,(q,q )) D (M M,N N ) and deg (q,q ) (f f ) = deg q (f) deg q (f ). Since (f f ) 1 (q,q ) = f 1 (q) f 1 (q ), this boils down to a consequence of elementary facts about determinants: if (p,p ) (f f ) 1 (q,q ), then f f is orientation preserving at (p,p ) if and only if f and f are either both orientation preserving or both orientation reversing at p and p respectively. The second property is a strong form of continuity. A continuous function f : C N with compact domain C M is degree admissible over q N if f 1 (q) C =. If this is the case, then there is a neighborhood U C N of the graph of f and a neighborhood V N \f( C) of q such that deg q (f ) = deg q (f ) whenever f,f : C N are smooth functions whose graphs are contained in U, q,q V, q is a regular value of f, and q is a regular value of q. We can now define deg q (f) to be the common value of deg q (f ) for such pairs (f,q ). Let D(M,N) be the set of pairs (f,q) in which f : C N is a continuous function with compact domain C M that is degree admissible over q N. The fully general form of degree theory asserts that (f,q) deg q (f) is the unique function from D(M,N) to Z such that: (D1) deg q (f) = 1 for all (f,q) D(M,N) such that f is smooth, f 1 (q) is a singleton {p}, and f is orientation preserving at p. (D2) deg q (f) = r i=1 deg q(f Ci ) whenever (f,q) D(M,N), the domain of f is C, and C 1,...,C r are pairwise disjoint compact subsets of U such that f 1 (q) C 1... C r \( C 1... C r ). (D3) If (f,q) D(M,N) and C is the domain of f, then there is a neighborhood U C N of the graph of f and a neighborhood V N \f( C) of q such that deg q (f ) = deg q (f ) whenever f,f : C N are continuous functions whose graphsarecontained in U and q,q V.
19 1.4. INDEX AND DEGREE The Fixed Point Index Although the degree can be applied to continuous functions, and even to convex valued correspondences, it is restricted to finite dimensional manifolds. For such spaces the fixed point index is merely a reformulation of the degree. Its application to general equilibrium theory was initiated by Dierker (1972), and it figures in the analysis of the LemkeHowson algorithm of Shapley (1974). There is also a third variant of the underlying principle, for vector fields, that is developed in Chapter 15, and which is related to the theory of dynamical systems. Hofbauer (1990) applied the vector field index to dynamic issues in evolutionary stability, and Ritzberger (1994) applies it systematically to normal form game theory. However, it turns out that the fixed point index can be generalized much further, due to the fact that, when we are discussing fixed points, the domain and the range are the same. The general index is developed in three main stages. In order to encompass these stages in a single system of terminology and notation we take a rather abstract approach. Fix a metric space X. An index admissible correspondence for X is an upper semicontinuous correspondence F : C X, where C X is compact, that has no fixed points in C. An index base for X is a set I of index admissible correspondences such that: (a) f I whenever C X is compact and f : C X is an index admissible continuous function; (b) F D I whenever F : C X is an element of I, D C is compact, and F D is index admissible. Definition Let I be an index base for X. An index for I is a function Λ X : I Z satisfying: (I1) (Normalization) If c : C X is a constant function whose value is an element of intc, then Λ X (c) = 1. (I2) (Additivity) If F : C X is an element of I, C 1,...,C r are pairwise disjoint compact subsets of C, and FP(F) intc 1... intc r, then Λ X (F) = i Λ X (F Ci ). (I3) (Continuity) For each element F : C X of I there is a neighborhood U C X of the graph of F such that Λ X (ˆF) = Λ X (F) for every ˆF I whose graph is contained in U. For each m = 0,1,2,... an index base for R m is given by letting I m be the set of index admissible continuous functions f : C R m. Of course (I1)(I3) parallel (D1)(D3), and it is not hard to show that there is a unique index Λ R m for I m given by Λ R m(f) = deg 0 (Id C f). We now extend our framework to encompass multiple spaces. An index scope S consists of a class of metric spaces S S and an index base I S (X) for each X S S such that
20 16 CHAPTER 1. INTRODUCTION AND SUMMARY (a) S S contains X X whenever X,X S S ; (b) F F I S (X X ) whenever X,X S S, F I S (X), and F I S (X ). These conditions are imposed in order to express a property of the index that is inherited from the multiplicative property of the degree for cartesian products. The index also has an additional property that has no analogue in degree theory. Suppose that C R m and C R m are compact, g : C C and g : C C are continuous, and g g and g g are index admissible. Then Λ R m( g g) = Λ R m(g g). When g and g are smooth and the fixed points in question are regular, this boils down to a highly nontrivial fact of linear algebra (Proposition ) that was unknown prior to the development of this aspect of index theory. This property turns out to be the key to moving the index up to a much higher level of generality, but before we can explain this we need to extend the setup a bit, allowing for the possibility that the images of g and g are not contained in C and C, but that there are compact sets D C and D C with g(d) C and g( D) C that contain the relevant sets of fixed points. Definition A commutativity configuration is a tuple (X,C,D,g, ˆX,Ĉ, ˆD,ĝ) where X and ˆX are metric spaces and: (a) D C X, ˆD Ĉ ˆX, and C, Ĉ, D, and ˆD are compact; (b) g C(C, ˆX) and ĝ C(Ĉ,X) with g(d) intĉ and ĝ(ˆd) intc; (c) ĝ g D and g ĝ ˆD are index admissible; (d) g(fp(ĝ g D )) = FP(g ĝ ˆD ). After all these preparations we can finally describe the heart of the matter. Definition An index for an index scope S is a specification of an index Λ X for each X S S such that: (I4) (Commutativity) If (X,C,D,g, ˆX,Ĉ, ˆD,ĝ) is a commutativity configuration with X, ˆX S S, (D,ĝ g D ) I S (X), and (ˆD,g ĝ ˆD) I S ( ˆX), then The index is said to be multiplicative if: Λ X (ĝ g D ) = Λ ˆX (g ĝ ˆD ). (M) (Multiplication) If X,X S S, F I S (X), and F I S (X ), then Λ X X (F F ) = Λ X (F) Λ X(F ).
Fixed Point Theory. With 14 Illustrations. %1 Springer
Andrzej Granas James Dugundji Fixed Point Theory With 14 Illustrations %1 Springer Contents Preface vii 0. Introduction 1 1. Fixed Point Spaces 1 2. Forming New Fixed Point Spaces from Old 3 3. Topological
More informationFIXED POINT SETS OF FIBERPRESERVING MAPS
FIXED POINT SETS OF FIBERPRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 email: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics
More information1 Local Brouwer degree
1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationTOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable
More informationORIENTATIONS. Contents
ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit nsimplex which
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationTOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms
More informationSets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use
More informationIn mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)
Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematics
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationA CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE
A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semilocally simply
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationLecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS
1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationLecture 11,2. Definition. Sets A and B have the same cardinality if there is an invertible function f : A B.
Topology Summary This is a summary of the results discussed in lectures. These notes will be gradually growing as the time passes by. Most of the time proofs will not be given. You are supposed to be able
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationFIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3
FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationThe Tangent Bundle. Jimmie Lawson Department of Mathematics Louisiana State University. Spring, 2006
The Tangent Bundle Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2006 1 The Tangent Bundle on R n The tangent bundle gives a manifold structure to the set of tangent vectors
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationPersuasion by Cheap Talk  Online Appendix
Persuasion by Cheap Talk  Online Appendix By ARCHISHMAN CHAKRABORTY AND RICK HARBAUGH Online appendix to Persuasion by Cheap Talk, American Economic Review Our results in the main text concern the case
More informationFixed Point Theorems in Topology and Geometry
Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSingular fibers of stable maps and signatures of 4 manifolds
359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationQuantum manifolds. Manuel Hohmann
Quantum manifolds Manuel Hohmann Zentrum für Mathematische Physik und II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Abstract. We propose a mathematical
More information6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium
6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline PricingCongestion Game Example Existence of a Mixed
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationominimality and Uniformity in n 1 Graphs
ominimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationThe Basics of Graphical Models
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. FiniteDimensional
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationVector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More information15 Limit sets. Lyapunov functions
15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationSPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM
SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More information4. CLASSES OF RINGS 4.1. Classes of Rings class operator Aclosed Example 1: product Example 2:
4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets
More informationCOBORDISM IN ALGEBRA AND TOPOLOGY
COBORDISM IN ALGEBRA AND TOPOLOGY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ aar Dedicated to Robert Switzer and Desmond Sheiham Göttingen, 13th May, 2005 1 Cobordism There is a cobordism equivalence
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationThe BanachTarski Paradox
University of Oslo MAT2 Project The BanachTarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the BanachTarski paradox states that for any ball in R, it is possible
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationINTRODUCTION TO TOPOLOGY
INTRODUCTION TO TOPOLOGY ALEX KÜRONYA In preparation January 24, 2010 Contents 1. Basic concepts 1 2. Constructing topologies 13 2.1. Subspace topology 13 2.2. Local properties 18 2.3. Product topology
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More informationComments on Quotient Spaces and Quotient Maps
22M:132 Fall 07 J. Simon Comments on Quotient Spaces and Quotient Maps There are many situations in topology where we build a topological space by starting with some (often simpler) space[s] and doing
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More information