Lecture Notes on Geometry and Topology Kevin Zhou [email protected] These notes cover geometry and topology in physics. They focus on how the mathematics is applied, in the context of particle physics and condensed matter, with little emphasis on rigorous proofs. For instance, no point-set topology is developed or assumed. The primary sources were: • Nakahara, Geometry, Topology, and Physics. The standard textbook on the subject for physi- cists. Covers the usual material (simplicial homology, homotopy groups, de Rham cohomology, bundles), with extra chapters on advanced subjects like characteristic classes and index theo- rems. The material is developed clearly, but there are many errata, and the most comprehensive list is available here. At MIT I co-led an undergraduate reading group on this book with Matt DeCross, who wrote problem sets available here. • Robert Littlejon's Physics 250 notes. These notes primarily follow Nakahara, with some extra physical applications. They are generally more careful and precise. • Nash and Sen, Topology and Geometry for Physicists. A shorter alternative to Nakahara which covers the usual material from a slightly more mathematical perspective. For instance, topological spaces are defined, and excision and long exact sequences are used. However, many propositions are left unproven, and some purported proofs are invalid. • Schutz, Geometrical Methods of Mathematical Physics. A basic introduction to differential geometry that focuses on differential forms. • Baez, Gauge Fields, Knots, and Gravity. Covers differential geometry and fiber bundles as applied in gauge theory. Like Nash and Sen, it has a \math-style" presentation, but not rigorous proofs. Also contains neat applications to Chern-Simons theory and knot theory. • Jeevanjee, An Introduction to Tensors and Group Theory for Physicists. A pedagogical introductory-level book that could serve as a prerequisite for these notes. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Preliminaries 3 1.1 Constructing Spaces....................................3 1.2 Topological Invariants...................................4 1.3 Euler Characteristic....................................5 2 Simplicial Homology8 2.1 Simplicial Complexes....................................8 2.2 Simplicial Homology.................................... 10 2.3 Computation of Homology Groups............................ 12 2.4 Properties of Homology Groups.............................. 15 3 Homotopy Groups 18 3.1 The Fundamental Group.................................. 18 3.2 Examples of Fundamental Groups............................ 19 3.3 Fundamental Groups of Polyhedra............................ 23 3.4 Higher Homotopy Groups................................. 24 3.5 Topological Defects..................................... 27 4 Manifolds 31 4.1 Smooth Manifolds..................................... 31 4.2 The Tangent Space..................................... 32 4.3 The Cotangent Space.................................... 35 4.4 Pushforward and Pullback................................. 37 4.5 Vector Fields and Flows.................................. 39 5 Lie Theory 42 5.1 The Lie Derivative..................................... 42 5.2 Frobenius' Theorem.................................... 44 5.3 Lie Groups and Lie Algebras............................... 46 5.4 Matrix Groups....................................... 51 6 Differential Forms 54 6.1 Geometric Intuition.................................... 54 6.2 Operations on Forms.................................... 56 6.3 Volume Forms........................................ 58 6.4 Duals of Forms....................................... 60 6.5 The Exterior Derivative.................................. 63 6.6 Stokes' Theorem...................................... 66 6.7 de Rham Cohomology................................... 68 6.8 Physical Applications................................... 72 7 Fiber Bundles 75 7.1 Motivation......................................... 75 7.2 Definitions.......................................... 79 7.3 Principal Bundles...................................... 82 7.4 Connections on Fiber Bundles............................... 85 3 1. Preliminaries 1 Preliminaries 1.1 Constructing Spaces Before diving into the formal definitions, we'll look at some at examples of spaces with nontrivial topology. Informally, a `space' X is some set of points, such as the plane. Let ∼ be an equivalence relation. Then the quotient space X= ∼ is the result of `gluing together' all points which are equivalent under ∼ . Example. Consider the real line R, and let x ∼ y if x − y is an integer. Then the quotient space is the unit interval [0; 1] with edges identified, which is topologically the circle S1. 2 Example. Consider a closed unit square in the plane R . We can identify various sides to get different results. • Identifying two opposite sides in the same direction gives a cylinder, while identifying them in the opposite direction gives a Mobius strip; the latter is an example of a nonorientable surface. • Identifying both pairs of opposite sides in the same direction gives the torus T 2. • Identifying one pair of sides in the opposite direction gives the Klein bottle; doing this to both gives the projective plane RP 2. Both the Klein bottle and the projective plane (shown above) are non-orientable, and cannot be 3 embedded in R without self-interaction. Another construction of the projective plane is to take the two-sphere S2 and identify opposite points. This is equivalent to taking the closed northern hemisphere of the sphere and identifying opposite edges of the equator; deforming the hemisphere into a square gives our construction. Example. Identifying all points on the boundary of the closed unit disc D2 yields the sphere S2, as shown below. More generally, identifying the boundary of Dn (i.e. constructing Dn=Sn−1) yields Sn. n n+1 Example. The n-dimensional real projective space RP is the set of nonzero vectors in R under the equivalence relation x ∼ y if x = cy for real nonzero c. That is, it is the set of two-sided lines in one higher dimension. 4 1. Preliminaries 2 This definition differs from our earlier construction of RP , but we can link it back. Every two- n n sided line intersects a unit sphere in two antipodal points, so RP is S with antipodes identified. Equivalently, it is the closed northern hemisphere of Sn with opposite points on the equator identified. But this is just Dn with opposite points on the boundary identified. For the case n = 2, deforming the disc to a square gives our original construction. Example. States (kets) in quantum mechanics live in a complex vector space, but they must be normalized and are insensitive to overall phase. Therefore kets are actually equivalence classes, [j i] = fcj i j c 2 C; c 6= 0g: We call [j i] a ray in the Hilbert space; the set of rays is a complex projective space. Example. Polarizations of light. The electric field of a monochromatic light wave propagating in the z direction may be written as h i E = Re aei(kz−!t) 2 where a is a complex polarization living in C . Then the space of polarization states is naively 2 4 C = R . If we only care about the direction of the electric field, not its magnitude, we identify a with ca for any real c > 0 and ignore a = 0. The resulting quotient space is S3. However, there is a further ambiguity in the phase of the wave. By changing the origin of time, we can move phase factors between a and the ei(kz−!t) factor. We thus identify a and eiαa. The resulting quotient space S3= ∼ is S2, and this construction is called the Hopf fibration. 1.2 Topological Invariants Definition. Let X1 and X2 be topological spaces. A continuous map f : X1 ! X2 with a continuous inverse is called a homeomorphism. If there exists a homeomorphism between X1 and X2, then the two spaces are homeomorphic. Homeomorphism is an equivalence relation, and we will use it to identify two topological spaces as \the same". Intuitively, if two spaces are homeomorphic, the spaces may be \deformed into each other", though this comes with some caveats. 3 Example. A ring in R may be considered a topological space by the subspace topology. However, the topological space consisting of two linked rings is homeomorphic to one containing two unlinked 3 rings, even though there exists no continuous map between the two in R . Topologically, the 3 embedding of the rings in R is irrelevant; the only important feature is that there are two of them. It is possible to detect the linking of the rings topologically, but this takes more work. For 3 example, if S is the set of points in the rings, we may compute the fundamental group of R n S. In theory, we could try to classify the equivalence classes of homeomorphism, but this is intractable. An easier goal is to define and compute topological invariants, i.e. properties of topological spaces which are invariant under homeomorphism. Then the invariants will give us a coarser, but hopefully still useful, set of equivalence classes. Invariants can be numbers, binary properties, or more generally algebraic objects such as groups. Connectedness, compactness, and the Hausdorff property are all topological invariants. Example. A closed interval [a; b] is not homeomorphic to an open interval (a; b), since only the former is compact. However, (−π=2; π=2) is homeomorphic to (−∞; 1) by the tangent function, so boundedness is not a topological invariant. 5 1. Preliminaries 2 2 Example. The open disc D is homeomorphic