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Homotopy Theory and Characteristic Classes

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Homotopy Theory and Characteristic Classes Homotopy Theory and Characteristic Classes Stefania Mombelli, BSc Physics Proseminar in Theoretical Physics ETH Zürich, spring semester 2018 Organiser: Prof. Matthias Gaberdiel Supervisor: Dr. Blagoje Oblak Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 Contents 1 Abstract 2 2 Introduction 3 3 Homotopy Theory 3 3.1 Basic notions of topology . 3 3.2 Homotopy of mappings . 3 3.3 Homotopy groups . 5 3.3.1 The fundamental group π1(Y ) ............................. 5 3.3.2 Other homotopy groups πn(Y ) ............................. 6 3.3.3 Some examples . 7 3.4 Topological quantum numbers . 8 4 Gauge Fields as Dierential Forms 9 4.1 Basic notions of dierential geometry . 9 4.1.1 Fiber bundles . 10 4.1.2 Section and connection . 10 4.2 Gauge elds as dierential forms . 11 4.2.1 Abelian gauge theory . 11 4.2.2 Non-abelian gauge theory . 12 5 Characteristic Classes 12 5.1 Basic notions . 13 5.2 Characteristic Classes . 13 5.3 Chern Classes . 14 5.4 Chern numbers . 15 6 Conclusion 16 7 References 16 1 Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 1 Abstract The purpose of this proseminar was to nd a way to classify particular types of topological spaces called ber bundles. In order to do this, we examined dierent areas of topology and dierential geometry, namely homotopy theory and characteristic classes. First of all, we studied the denitions of homotopy of mappings and homotpy groups and we analyzed topological quantum numbers, which are important applications. Then, we made a short digression on dierential geometry and gauge theories, which allowed us to understand the concept of characteristic classes. We have seen how homotopy theory and characteristic classes are related with two examples, regarding the cases of an abelian gauge theory on the plane and of a non-abelian gauge theory on the Euclidean spacetime. We have seen how the classication of ber bundles by means of homotopy theory and characteristic classes leads to a quantization, which in some cases can also be measured physically. An example of this appears in the Quantum Hall Eect, which we have not discussed in detail. 2 Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 2 Introduction The topic which we are going to discuss in this report is one of the rst ones presented in the proseminar "Algebra, Topology and Group Theory in Physics". It constitutes an introduction to the mathematical concepts which are necessary to explain some physical phenomena analyzed in the other proseminars. We are going to refer to some of these applications without examining them in detail. The main subjects of this work are homotopy theory and characteristic classes, which are two areas of topology and dierential geometry. Our purpose is to use both these dierent and apparently unconnected theories to classify particular types of mathematical spaces, called ber bundles, and collect them into equivalence classes. In physics, this leads to the quantization of some observables associated with these spaces, as we will see. The report is divided into three sections. The rst one is about homotopy theory, which deals with continuous deformations of mappings and topological spaces. The second part contains an introduction in dierential geometry and gauge theories, which are necessary concepts in order to understand the last part of the work, which is about characteristic classes. In particular, we are interested in Chern classes, which are a type of characteristic classes, and we will see how these are closely related to homotopy theory. 3 Homotopy Theory In order to discuss homotopy theory, we rst need to understand some important concepts of topology. 3.1 Basic notions of topology Topology is an area of mathematics which deals with the properties of space which are preserved under continuous deformations. The most important notion in topology is the notion of topological space. The following denition is not mathematically rigorous, but it is sucient for the purposes of this work. Denition 3.1.1 (Topological space). A topological space is a mathematical space that allows for the denition of concepts such as continuity, connectedness, and convergence. In the following, we are going to deal with a particular type of mathematical spaces called manifols. Denition 3.1.2 (Manifold). A manifold is a topological space which locally looks like the Euclidean space. This means that each point on a n-dimensional manifold has a neighborhood which is homeomorphic to Rn. 3.2 Homotopy of mappings The starting point for dealing with homotopy theory is the following denition. Denition 3.2.1 (Homotopic mappings). Let X, Y be topological spaces and let f, g: X −! Y be two continuous mappings. f and g are said to be homotopic if there exists a continuous mapping h : X × [0; 1] −! Y (x; t) 7! y such that h(x; 0) = f(x) and h(x; 1) = g(x). This means that f is continuously deformable into g. Example 3.2.1. Let X = S1 and Y = R2 nf0g. Let f, g be two mappings from X to Y, as shown in gure 1 . Then f is homotopic to the constant map, while g is not. It follows that f and g are not homotopic. Statement 3.2.1. Homotopy is an equivalence relation. 3 Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 Figure 1: [2] Proof. • Symmetry: if f is homotopic to g, then g is homotopic to f since the ow of the parameter t can be reversed. • Transitivity: f homotopic to g and g homotopic to k implies that f is homotopic to k, since intervals can be adjoined and rescaled. • Reexivity: f is homotopic to itself, since we can choose h(x; t) = f(x) 8t 2 [0; 1]. The homotopy relation divides the set of continuous mappings from X to Y into equivalence classes, called homotopy classes, which we denote by { X,Y}. The denition of homotopic mappings is related to the concept of homotopic topological spaces. Denition 3.2.2 (Homotopic topological spaces). Let X, Y be topological spaces. X and Y are said do be homotopic if there exist mappings f : X −! Y and g : Y −! X such that f ◦ g : Y −! Y and g ◦ f : X −! X are homotopic to the identity. Example 3.2.2. The plane R2 is homotopic to a point P. In fact, we can construct the following map- pings: 2 2 f : R −! P and g : P −! R (x; y) 7! P P 7! (0; 0): The mapping f ◦ g : P −! P is the identity. The mapping g ◦ f : R2 −! R2 is homotopic to the identity. The homotopy is given by the mapping: 2 2 h : R × [0; 1] −! R ((x; y); t) 7! h((x; y); t) = (tx; ty): We now report a statement which allows us to analyze the homotopy classes {X,Y} in a simple way. Statement 3.2.2. Let X, Y1 and Y2 be topological spaces and let Y1 and Y2 be homotopic. Then there exists a bijection between {X,Y1} and {X,Y2}. Conversely, if X1, X2 and Y are topological spaces and X1 and X2 are homotopic, there exists a bijection between {X1,Y } and {X2,Y }. 4 Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 3.3 Homotopy groups An interesting case consists in considering homotopy classes of mappings from the n-sphere Sn to a topological space Y . In fact, in this particular case, it is possible to dene an operation between mappings which induces an operation between homotopy classes and which confers to the set of classes a group structure. In order to dene the group operation, the following denition is useful. Denition 3.3.1 (Based maps). Let X and Y be topological spaces and let x0 2 X and y0 2 Y be xed points. Maps f : X −! Y with f(x0) = y0 are called based maps with base points x0 and y0. We start from the case n = 1, that is we consider the homotopy classes of based maps from the circle 1 S to a space Y . The resulting group is called the fundamental group of Y and is denoted by π1(Y ). Then, we will generalize this construction to mappings from the spheres Sn of higher dimensions n ≥ 2 to Y . In this case, we use the notation πn(Y ). 3.3.1 The fundamental group π1(Y ) We can view the based mappings f from S1 to Y as mappings from the interval [0,1] to Y , such that f(0) = f(1) = y0. These maps are associated with paths in Y , beginning and ending at the point y0. We can dene an operation f · g between two mappings f and g by taking the concatenation of the corresponding paths. This means that we construct the path f · g by running rst along the path of f and then along the one of g, as shown in gure 2. Figure 2 Denition 3.3.2. Formally, the mapping f · g can be dened as 8 <f(2t); 0 ≤ t ≤ 1=2 (f · g)(t) = :g(2t − 1); 1=2 ≤ t ≤ 1: We now need to show that this operation is well dened, that is it induces an operation between the homotopy classes of the considered mappings. In order to do this, it is enough to show the following statement. 1 Statement 3.3.1. Consider based maps f, f', g and g' from S to Y, starting and ending at y0, such that f is homotopic to f' and g is homotopic to g'. Then f 0 · g0 is homotopic to f · g.
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