Math 635: Algebraic Topology III, Spring 2016

Transcription

1 Math 635: Algebraic Topology III, Spring 2016 Instructor: Nicholas Proudfoot Office: 322 Fenton Hall Office Hours: Monday and Tuesday 2:00-3:00 or by appointment. Text: We will use a number of different sources, as indicated in the course outline below. All of these sources can be found on-line, though I recommend having a hard copy of the book by Guillemin and Pollack [GP10]. You can probably borrow one from somebody who took Math 532 last winter. Exams: There will be one mid-term exam, not yet scheduled. We will also try to reschedule the final exam for a mutually convenient time. Homework: Homework assignments will be due at random times, and must be typed using LaTeX. Collaboration on homework assignments is allowed and encouraged, but you must write up your solutions independently. Grading: Your score in the class will be determined by your final exam (35%), mid-term exam (15%), and homework assignments (50%). The correspondence between numerical scores and letter grades will be determined at the end of the term. Here is some indication of what material I plan to cover, in what order, from what sources. I have also attempted to collect some good exercises. Probably the homework that I eventually assign will neither contain nor be contained in the set of problems mentioned below. 1 Manifolds 1.1 What s a manifold? We will take the material of [GP10, 1.1] as our basic definition of the category of smooth manifolds, and look at a bunch of examples. However, we will also discuss two alternative (and ultimately equivalent) definitions. The first is in terms of an abstract topological space equipped with a smooth atlas; this is what you d find (for example) on Wikipedia. This is not very different from the definition in [GP10]; the main difference is that Guillemin and Pollack don t want to assume that you know what a topological space is! This definition makes it easier to think about certain classes of examples, such as RP n, CP n, quotients of manifolds by free finite group actions, and covering spaces of manifolds. 1

2 The other definition, which is significantly more sophisticated, is given in terms of a topological space equipped with a sheaf of rings. A good reference for this is [Ten75, 4.3]. We won t do very much with this definition, but we will come back to it from time to time for your general edification. Exercises: 6, 9, 12, 16, and 17 from [GP10, 1.1]. 1.2 Derivatives We ll define tangent spaces and talk about the derivative of a map between manifolds, mostly following [GP10, ]. The main results here are the chain rule, the inverse function theorem, the local immersion theorem, the local submersion theorem, and the preimage theorem. This will allow us to give a lot more examples of manifolds. We ll also give the sophisticated sheafy definition of a tangent space. Exercises: 3, 6, and 8 from [GP10, 1.2], 9 from [GP10, 1.3], and 5, 6, 7, 8, 10, 11, 12, and 13 from [GP10, 1.4]. 1.3 Transversality Here we will cover [GP10, 1.5]. We will define what it means for two submanifolds to be transverse, and (more generally) what it means for a map to be transverse to a submanifold. We will show that the preimage of a submanifold along a transverse map is a manifold, generalizing the preimage theorem. Last, we will state (but not prove) the transversality homotopy theorem from [GP10, 2.3]. Exercises: 2, 4, 5, 6, 7, 8, 9, and 10 from [GP10, 1.5]. 2 Vector bundles I need to find some exercises for this material! 2.1 What s a vector bundle? We ll define vector bundles, following either Hatcher s other book [Hat, 1.1] or the Wikipedia article (which I like a little bit better). Emphasize transition functions as a nice way to think about things very concretely. The main motivating example will be the tangent bundle of a manifold. Also maybe the tautological line bundles on RP n and CP n. We can be very explicit about the transition functions in these examples! Define homomorphisms of vector bundles and sections of vector bundles, and explain that a section is the same as a homomorphism from the trivial line bundle. Also talk about defining a vector bundle in terms of its sheaf of sections. 2.2 Natural operations We ll start by going over natural operations on vector spaces: Hom, dual (special case of Hom), direct sum, quotient, tensor product, symmetric power, exterior power. (For the latter two, it can 2

3 be helpful to think about dual vector spaces.) In each case, we ll emphasize the functoriality and work out explicitly what happens at the level of matrices. The most important example will be the top exterior power, which replaces a matrix by its determinant. Next, we ll note that we can do all of these things in families, that is, for vector bundles! Emphasize the perspective of transition functions to be very concrete. Observe that line bundles form a group under tensor product. Also talk about the pullback of a vector bundle along a map. The most important new examples that we will obtain are normal bundles to submanifolds. In particular, we ll state the tubular neighborhood theorem; see the exercises of [GP10, 2.3] for an indication of the proof. We ll want a few lemmas for later: the normal bundle to a transverse intersection of submanifolds is the direct sum of the restrictions of the normal bundles; the pullback of the normal bundle of a submanifold along a map transverse to that submanifold is the normal bundle of the preimage; homotopic maps have isomorphic pullbacks [Hat, 1.6]. 2.3 Orientations We ll review the notion of an orientation of a vector space, which is the same as a nonzero element (up to positive scalar) of the top exterior power. We ll generalize this to define an orientation of a vector bundle. I ll try to explain why this definition is equivalent to that in [Hat02, 4.D]. The most important special case is the tangent bundle to a manifold. In particular, we ll want to see that an orientation of the tangent bundle is equivalent to an orientation of the manifold as defined in [Hat02, 3.3]. I ll want to emphasize two lemmas: the transverse intersection of two cooriented submanifolds is naturally cooriented, and the transverse intersection of a cooriented submanifold with an oriented submanifold is naturally oriented. 3 Homology and cohomology classes of submanifolds 3.1 Homology classes We ll define the fundamental homology class of a nonempty, compact, connected, oriented submanifold, following [Hat02, 3.3]. We will also do the relative version (for manifolds with boundary), which Hatcher treats at the same time. Interesting (and later useful) tidbit: given any (homogeneous) homology class on X, some multiple of it is represented by a map from a compact oriented manifold to X. (I haven t been able to find a statement as precise as I would like, but see and references therein.) Exercises: 2-13, from [Hat02, 3.3]. 3

4 3.2 Cohomology classes We will begin by defining the Thom class of an oriented vector bundle, following [Hat02, 4.D]. Applying this construction to the normal bundle to a cooriented submanifold, we obtain the cohomology class of the submanifold. (There may be some subtlety about the lack of dependence on the choice of tubular neighborhood and the indentification of the tubular neighborhood with the total space of the normal bundle.) For motivation, we ll emphasize the intersection pairing between homology and cohomology. We will also show that this construction behaves nicely under pullback and transverse intersection. 3.3 Cap product We will define the (relative) cap product for arbitrary topological spaces, following the exposition in [Hat02, 3.3]. Using this, we can prove that cap product is given by transverse intersection [Hut, 4.1 & 4.2]. The fact that the pairing between homology and cohomology is given by transverse intersection is a special case. Over Q or Z 2, we can use our discussion of representability of homology classes to show that the cohomology class of a boundary is trivial. At this point, we can look at lots of nice examples! These should include C, projective spaces, maybe flags in C 3, maybe the Hirzebruch surface. Also simple hyperplane arrangements (defined over R for simplicity), which should go something like this: 1. Define the classes e 1,..., e n by pulling back from C. 2. Show by intersecting that every dependent set monomial is zero. 3. Let R A be the quotient of Λ Z [e 1,..., e n ] by the dependent set monomials, which maps to H (M A ). Prove injectivity by pulling back to the neighborhood of a vertex. 4. Consider the LES H k+1 (M A ) H k+1 (M A ) H k (M A ). By induction, we have H k+1 (M A ) = R k+1 A, which clearly injects into R k+1 A Hk+1 (M A ), hence the LES splits into short exact sequences. 5. Now do a dimension count to show that H (M A ) = R A. 4 Poincaré duality Now that we have a better geometric understanding of homology and cohomology, we ll go back and prove all of the duality theorems in [Hat02, 3.3]. We won t use any of the geometric stuff in the proofs, but it will motivate everything well! Exercises: from [Hat02, 3.3]. 4

5 5 De Rham cohomology We will discuss differential forms and de Rham cohomology, primarily following [GP10, 4]. Unfortunately, I don t think that we will have the machinery to prove the main theorem, which is that the de Rham cohomology ring is canonically isomorphic to the singular cohomology ring. But we will at least be able to show that you can evaluate a de Rham cohomology class on a compact oriented submanifold to get a number, and that this number vanishes when the submanifold is a boundary. Combined with the Thom representability result, we might be able to prove that de Rham cohomology is additively isomorphic to singular cohomology. If time permits, we will reinterpret many of our results in the context of de Rham cohomology, following [BT82, 5-7]. As an application of de Rham cohomology, we may spend some time looking at cohomology rings of complements of central hyperplane arrangements. References [BT82] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, [GP10] Victor Guillemin and Alan Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010, Reprint of the 1974 original. [Hat] Allen Hatcher, Vector bundles and K-theory. [Hat02], Algebraic topology, Cambridge University Press, Cambridge, [Hut] Michael Hutchings, Cup product and intersections, hutching/teach/215b- 2011/cup.pdf. [Ten75] B. R. Tennison, Sheaf theory, Cambridge University Press, Cambridge, England-New York- Melbourne, 1975, London Mathematical Society Lecture Note Series, No