Module Content

On successful completion of this module you should be able to:

• Establish the invariance of the Euler characteristic of a sphere, and compute simple Euler integrals.
• Give the definitions of a topological space and a continuous map between topological spaces, and provide examples.
• Understand connectedness and compactness as topological invariants.
• Understand the concept of homeomorphism, and use topological invariants to prove that certain spaces are not homeomorphic.
• Construct new topological spaces using the subspace and quotient constructions.
• Understand and represent simplicial complexes and triangulated spaces.
• Understand homotopy equivalence, and informally explain why the Euler characteristic is a homotopy invariant of a triangulated space.
• Prove the fundamental theorem of algebra, Perron’s theorem, Brouwer’s Fixed Point Theorem.
• Understand John Nash’s proof of the existence of Nash equilibria in game theory.
• Understand the basic idea behind topological data analysis.

Module Coordinates

• Lecturer: Graham Ellis
• Lectures: take place at 12pm Monday in the McMunn Theater and 12pm Wednesday in the McMunn Theater.
• Tutorials: take place at 2pm Wednesday in AC201 and 6pm Wednesday in AM150. The tutor is Kelvin Killeen.
• Recomended text: The lectures will be loosely based on the first five chapters of the text: Basic Topology by M.A. Armstrong, Undergraduate Texts in Mathematics, Springer-Verlag. The book emphasizes the geometric motivations for topology and I recommend that you take a look at it. There are some copies in the library. The text: Topology and Groupoids by R. Brown is also a great book which covers similar material and should be consulted too. It is available online here.
• Problem sheet: available here.
• Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA342 Topology pages. Blackboard will also be used for announcements and for posting grades.

Module Assessment

The end of semester exam will count for 70% of the assessment. The continuous assessment will count for 30% of the assessment.

The continuous assessment will consist of three equally weighted in-class tests based on the homework/tutorial problems. Each test will consist of questions taken verbatim from the homework/tutorial sheet. Many questions on the end of semester exam will also be taken, almost verbatim, from the homework/tutorial sheet too.

Supplementary Material and News

Why is Topology MA342 relevant to Maths students?

Topology can be fun. It is also a major branch of mathematics, as demonstrated by the number of Fields Medals awarded to topologists such as Atiyah, Donaldson, Freedman, Jones, Milnor, Mumford, Novikov, Perelman, Quillen, Serre, Smale, Thom, Thurston, Voevodsky ... . The module MA342 tries to give students a taste of this vast subject.

Why is Topology MA342 relevant to Computer Science students?

In the last decade or so, topologists have been trying to harness the power of modern computers to apply topological ideas to problems in science and engineering. The aim is to use the deformation invariant notions of topology to provide qualitative answers to problems; see, for instance, details of the research network on Applied Computational Algebraic Topology . The module MA342 tries to hint at these applications through a discussion of Euler characteristics of digital images and Euler integration in sensor networks.

Why is Topology MA342 relevant to Financial Maths & Economics Students?

Fixed point theorems play an important role in theoretical economics; see, for instance, the textbook Fixed point theorems with applications to economics. The module MA342 provides the outline of a proof of Brouwer's fixed point theorem and an explanation of how Brouwer's theorem can be used to prove the existence of Nash equilibria. This latter notion is due to the mathematician John Nash who was awarded the Nobel Prize for Economics for his work in this area.

Why is Topology MA342 relevant to Mathematics & Education Students?

Much of school mathematics focuses on procedural tasks: teach children the procedures for calculating answers to problems and then test their ability to do mathematics by asking them a range of problems to which the procedures can be applied. The core Maths modules in the Mathematics & Education BA programme also tend to focus to a large extent on procedural mathematics: evaluate a multiple integral; evaluate a complex integral, calculate the inverse of a matrix; determine a probability using Bayes' Rule; decipher an encrypted message by first using Euclid's algorithm to solve a system of equations; use differentiation to calculate the maximum/minimum value of some quantity; ... .

Project Maths has been introduced into schools with the noble aim of complementing childrens' procedural knowledge of mathematics with a strong conceptual knowledge. One difficulty facing teachers of Project Maths is: how can a child's conceptual knowledge of a topic be developed, and how can it be reliably assessed?

The MA342 module is primarily concerned with developing students' conceptual knowledge of a particular area of mathematics. Even though topology, per se, is unlikely to enter into the Project Maths curriculum in the near future, the module should give students some ideas for developing and assessing conceptual mathematics.

Clicker opinion polling may be used in some lectures.

Lecture Notes

 Lecture Notes Lecture Summaries 1 Explained that topology is the study of those properties of a space that remain unchanged through a "continuous deformation" of the space, but I waived my hands a bit too much when using the term "continuous deformation". In later lectures we'll see that hand waiving can be replaced my mathematically precise definitions. I then defined the Euler characteristic (=V-E+F) of a surface such as the surface of Mars, and "observed" that this number is a topological property of the surface. I used the Euler characteristic to count the number of pentagons in a soccer ball, the number of pentagons in a fullerene molecule, and the number of pentagons in a Buckminster Fuller dome (assuming that the basement of the dome completes the sphere). Here are some slides used during the lecture. If you are interested in applications of topology to finance and data science take a look at the Ayasdi company website. 2 Defined the Euler integral of an integer valued weight function w:X --> Z defined on a planar region X covered by a collection of (closed) regions Ui. The weight function of main interest is defined by w(x)=|{i : x is in Ui}| (so be careful of the typo in the slides relating to this definition). Explained how a Texas farmer could use the Euler integral to count cows on the ranch. This lecture is based on the recent research article: Target enumeration via Euler characteristic integrals by Yuliy Barishnikov (Bell Labs, New Jersey) and Robert Christ (University of Pennsylvania). Unfortunately, the method could equally well be used to count tanks in a battle field in the ongoing fight against the axis of evil. 3 Started by emphasizing that, in our "proof" that the Euler characteristic of the 2-sphere S2 is V-E+F=2, we used the fact that a simple closed curve on the 2-sphere cuts the 2-sphere into two pieces. Then observed that a simple closed curve on a möbius strip or on a torus does not necessarily cut the space into two pieces. So we should really justify (i.e. prove) that a simple closed curve on the 2-sphere always cuts it into two pieces. Gave some motivation for a more precise approach to topology and stated the Jordan Curve Theorem. Explained that the next few lectures would focus on obtaining a good understanding of the statement of this theorem and on the main features in its proof. Ended by giving the definition of a topological space. Next lecture we'll consider lots of examples aimed at reinforcing our understanding of this definition. 4 Gave the definition of a topological space, and of a connected topological space, and illustrated these two definitions with examples. 5 Gave the definition of a topological subspace and of a connected component of a space. 6 Began with a video about topology and data analysis. Then gave a brief introduction to topological data analysis. In the last ten minutes the definition of a continuous function between topological spaces was given. 7 Gave the definitions of continuous function and homeomorphism, and gave some examples. In Topology two spaces are considered to be "the same" precisely when they are homeomorphic. A property is said to be topological if, whenever some space X possesses the property then so too do all spaces Y that are homeomorphic to X. (So, for instance, the "sum of the interior angles" is not a topological property of a polygon since it is 180 degrees for a triangle, and 360 degrees for a quadrangle, and yet a triangle is homeomorphic to a quadrangle.) I didn't have time to show the classic example of a homeomorphism between a doughnut and a coffee mug so please take a look at it on-line. 8 Explained that the real interval (-1,1) is homeomorphic to the real line. Proved that "connectedness" is a topological property, and used this property to prove that the real line is NOT homeomorphic to the real plane. Finished up by justifying the use of the term "compact space" in favour of "finite space" or "bounded space", and explained one reason why we'd like to prove that "compactness" is a topological property. The precise definition of "compactness" will be given in the next lecture. 9 First in-class test. Monday 11 Februrary. 10 Gave the definition of compactness. Showed that the real line R is not compact. Showed that compactness is a topological property. Showed that the subspace [0,1] of the real line IS compact. Hence the real line is not homeomorphic to the interval [0,1]. 11 Started to construct a surjective continuous function f:=[0,1] ---> Solid triangle from the unit interval of the real line to an equilateral triangular region of the plane. The function f is constructed as the limit of a sequence of functions f1, f2, ... . Next lecture will see why f is continuous and surjective. 12 Explained what a Cauchy sequence is, and then stated that every Cauchy sequence in Euclidean space En converges. Used this fact to see that our function f:[0,1] ---> triangular region is well-defined. Continuity of f is straightforward. As preparation for the proof that f is surjective the notions of closed subset and accumulation point were introduced. Ended up with a proof that a subset of a topological space is closed if, and only if, it contains all its accumulation points. 13 In order to prove that our Peano curve f:[0,1] ---> Δ is surjective we introduced the notion of a Hausdorff topological space and proved that in such a space "compact" implies "closed". (Since [0,1] is compact we get that f([0,1]) is compact. Since Δ is Hausdorff we get that f([0,1]) is closed. So f([0,1]) contains all its accumulation points. Thus f is surjective since, by construction, every point in Δ is an accumulation point of f([0,1]).) 14 Introduced the notion of a simplicial complex. Defined a triangulation of a topological space X to consist of a simplicial complex K and a homeomorphism h:|K| ---> X. 15 Defined the Euler characteristix χ(X) of a space X to be the Euler characteristic χ(K) of any simplicial complex K with |K| homeomorphic to X. Stated the theorem that is needed to make sure this definition is well-defined. Gave some triangulations of a circle, torus, Möbius strip, 2-sphere and computed the Euler characteristic of these spaces from the triangulation. Talked briefly about the Hauptvermutung. 16 Introduced the notion of "homotopy" between maps. Showed that any two maps X--->Y are homotopic when Y is a convex subset of Euclidean space. For given spaces X and Y we proved that homotopy is an equivalence relation on the collection of maps X--->Y. 17 Second Class Test: 11 March 18 Defined what is means for two spaces to be homotopy equivalent. Showed that any two homeomorphic spaces are homotopy equivalent. Also showed that the space C\{0} of non-zero complex numbers is homotopy equivalent to a circle. Stated a major theorem: Homotopy equivalent spaces have the same Euler characteristic. Illustrated this theorem with a few examples. Stated Brouwer's fixed-point theorem: any continuous function f:Δn ---> Delta;n has a fixed point. Ended up showing John Nash's PhD thesis in which he uses Brouwer's theorem to prove the existence of (what are now called) miched Nash equilibria. 19 St Patrick's Day + 1 20 Stated, illustrated, and proved Brouwer's fixed point theorem. Then used it to show that any nxn matrix of positive real numbers has at least one eignevector, and that this eigenvector can be chosen so that its entries are all non-negative. Then showed a Holywood clip on Nash Equilibria which includes a shot of John Nash's PhD thesis in which he uses Brouwer's theorem to prove "the existence in any game of at least one equilibrium point". 21 Introduced the fundamental group of a space. Indicated why the fundamental group of the circle S1 is isomorphic to the additive group of the integers Z. This isomorphism was used to define the winding number of an arbitrary map f:S1 ---> S1. 22 Third class test: 27 March 23 Used the fundamental group of a circle to prove the Fundamental Theorem of Algebra. Then introduced the notions of (i) an n player game and (ii) a pure Nash equilibrium. We saw two examples, one with pure Nash equilibria and the other without any pure Nash equilibrium. 24 Defined a mixed Nash equilibrium and stated Nash's Theorem: in any game with finitely many players with finite strategy sets there exists a mixed Nash equilibrium. Gave an outline proof of this theorem using Brouwer's fixed point theorem.