Math 215a: Algebraic topology

Instructor

About the course

In the first semester we will begin to study the essential "core" material of algebraic topology: homotopy groups, homology, and cohomology, and interesting examples and applications thereof.

In the second semester we will finish whatever is left over from the first semester, and then study a selection of more advanced, but still very important topics. Possible topics include but are not limited to bundles and characteristic classes, Morse theory, and spectral sequences.

Textbook

Prerequisites

Lecture summaries and plans

• Introduction.

  • (8/26) In the first lecture, I gave a survey of what algebraic topology is about, and some of the interesting problems in topology (both solved and unsolved) which motivate me as a topologist and which algebraic topology can help solve.

• The fundamental group and covering spaces.

  • (8/28)
    • We defined the fundamental group, and computed it for R^n, S^1 (by lifting paths to R), products, and S^n for n>1 (by using smooth approximation to miss a point, or by cutting S^n into two simply connected pieces whose intersection is connected), compare section 1.1 of the book.
    • For a more interesting example, we introduced the braid group (which is not discussed in the book).
  • (9/2)
    • We did some applications of the fundamental group of the circle, namely the Brouwer fixed point theorem and Borsuk-Ulam theorem in 2 dimensions, and the fundamental theorem of algebra. This is in section 1.1. (We will generalize the first two theorems to higher dimensions later after developing more machinery.)
    • Then I tried to give the basic idea, without proving everything, of:
      • the Seifert-Van Kampen theorem for the fundamental group of the union of two spaces,
      • the fact that the fundamental group of a CW complex with one 0-cell has a presentation with one generator for each 1-cell and one relation for each 2-cell.
      These are fully proved in section 1.2 of the book. (For details on the topology of CW complexes, see the Appendix in the book.)
  • (9/4)
    • We discussed the following topics from from sections 1.1 and 1.2 of the book:
      • We explained to what extent the fundamental group depends on the base point.
      • We showed that the fundamental group is invariant under deformation retracts and did some examples.
      • We gave more details about CW complexes and their fundamental groups.
    • Then we computed some examples of fundamental groups of link complements and showed that the Hopf link and the Borromean rings are linked and that the trefoil knot is knotted.
  • (9/9)
    • We completed the discussion of sections 1.1 and 1.2 by showing that a homotopy equivalence induces an isomorphism on fundamental groups.
    • Then we introduced covering spaces, which are covered (pun) in section 1.3.
      • We discussed many examples, including the relation between finite covers of S^1 and permutations, coverings of the torus, and, going slightly off-topic, the covering of any genus g>1 surface by the hyperbolic plane, the non-example of branched coverings of Riemann surfaces, and the canonical double covering of any manifold by an oriented manifold.
      • Then we proved the fundamental lemma on lifting paths to a covering space.
  • (9/11)
    • We discussed the map from the fundamental group of a covering space in the fundamental group of the base.
    • We proved part of the correspondence between path connected covering spaces of a reasonable space and subgroups of its fundamental group.
    • We briefly mentioned the analogy between covering spaces and Galois theory (which is not in the book).
  • (9/16)
    • We proved the Lifting Criterion and used this to complete the proof of the correspondence between path connected covering spaces and subgroups of the fundamental group.
    • We proved that any subgroup of a free group is free.
    • As a preview, we introduced higher homotopy groups and discussed how they are hard to compute, so we will study homology theory first. Note that regarding the "algebraic" aspect of algebraic topology, so far we have seen a lot of group theory; next there will be less group theory and more linear algebra.

• Homology.

  • (9/18) We defined the simplicial homology of a Delta-complex, and discussed some examples and intuition. See the first few pages of Chapter 2.
  • (9/23)
    • We defined the singular homology of a topological space. We computed H_0 of any space.
    • We computed the homology of star-shaped subsets of R^n by taking cones over simplices. (This is not in the book. Note also that this argument easily extends to compute the homology of any contractible space.)
    • We began to explain the relation between pi_1 and H_1 by defining a canonical homomorphism from the abelianization of pi_1 to H_1.
  • (9/25)
    • For a path connected space, we defined a homomorphism from H_1 to the abelianization of pi_1, which is the inverse of the map defined in the previous lecture. (This is not in the book, which takes a different approach to prove that the latter map is an isomorphism for a path connected space. My approach avoids the classification of surfaces and can be generalized to prove the Hurewicz theorem relating pi_k and H_k.)
    • We proved that homology is a homotopy invariant. Unlike the book, we used the handy trick of "acyclic models" to avoid explicitly constructing a "triangulation" of a prism, by using homology theory to show that a "triangulation" with the desired properties must exist.
    • We then summarized the algebra involved by defining chain complex, chain map, and chain homotopy. We have associated to each topological space a chain complex (with an associated homology), to each continuous map a chain map (which induces a map on homology), and to each homotopy a chain homotopy between the corresponding chain maps (which shows that these chain maps induce the same map on homology).
  • (9/30)
    • Discussed exact sequences.
    • Stated Mayer-Vietoris sequence, used it to compute homology of S^n. (See section 2.2.)
    • Discussed how a short exact sequence of chain complexes induces a long exact sequence in homology groups. (Left most of the "diagram chasing" as an exercise. This is also in section 2.1.)
    • Proved Mayer-Vietoris assuming "subdivision lemma" (which is in section 2.1).
    • Discussed the connecting homomorphism in the Mayer-Vietoris sequence.
  • (10/2) We discussed the Mayer-Vietoris sequence in more detail and worked out some examples. The goal was to develop some intuition for how the abstract machinery discussed in the last lecture relates to concrete geometrical considerations.
  • (10/7) We introduced the following topics from section 2.1 of the book:
    • relative homology
    • the long exact sequences of a pair and a triple
    • excision
    • the relation between relative homology and absolute homology for "good pairs".
  • (10/9)
    • Tied up loose ends about excision from last time.
    • Proved the equivalence of simplicial and singular homology, in which the Five Lemma made its first appearance.
    • Introduced cellular homology of a CW-complex.
    This finishes our coverage of section 2.1 of the book (and we have also seen a bit of section 2.2).
  • (10/14)
    • Discussed the degree of maps from a sphere to itself (including some perspective not in the book, namely the differential topology definition of degree and a sketch of why degree defines an isomorphism from pi_n(S^n) to Z).
    • Proved the "hairy ball theorem".
    • Proved the isomorphism between singular and cellular homology of a CW-complex. Studied the example of complex and real projective space.
    (Most of the above is in section 2.2.)
  • (10/16)
    • Proved the generalized Jordan Curve Theorem. (See section 2.B.)
    • As an interesting example, discussed linking number in detail. (This is not in the book.)
  • (10/21)
    • Discussed Euler characteristic. (Section 2.2.)
    • Proved Lefschetz fixed point theorem for simplicial complexes. (Section 2.C.)
  • (10/23)
    • Homology with coefficients.
    • Tor.
  • (10/28)
    • More about Tor and the universal coefficient theorem.
    • Start of Kuenneth formula for the homology of the product of two spaces. (Note: I am doing the geometric part of this completely differently from the book, using acyclic models and proving the Eilenberg-Zilber theorem. For this approach see e.g. Geometry and Topology by Bredon. You can also read Hatcher's approach and see which you prefer.)
  • (10/30)
    • Finished Kuenneth formula.
    • Started cohomology. (See intro to chapter 3 and section 3.1.)

• Cohomology.

  • (11/4)
    Today's theme: What is cohomology?
    • Discussed simplicial cohomology of a surface.
    • Stated Poincare duality (will prove later).
    • Defined cup product. (See section 3.2.)
    • Stated the fact that Poincare duality intertwines cup product with intersection of submanifolds. I haven't found this in Hatcher, but I think it is the most important thing to understand about cup product, and we will discuss it a lot more.
  • (11/6)
    • Worked out the cohomology ring of projective spaces.
    • Used this to prove the Borsuk-Ulam theorem. (This is a different proof of Borsuk-Ulam than Hatcher's, which uses "transfer homomorphisms" instead.)
    • Discussed the cohomology ring of the product of two spaces, assuming that the cohomology module of one of the spaces is finite rank and free. Example: the cohomology ring of the n-torus is an exterior algebra with n degree 1 generators.
    • Outlined the proof that cohomology of a smooth manifold with real coefficients is isomorphic to de Rham cohomology (for those of you who know what that is).
  • (11/13) Started explaining the important ideas in the proof of Poincare duality, for which purpose we discussed:
    • The R-orientation bundle, and the identification between sections over a compact set A and H_n(X,X-A;R).
    • The relative Mayer-Vietoris sequence.
    • Compactly supported cohomology.
    • More fun with direct limits!
  • (11/18) Lecture cancelled. (I am going to a topology conference.) To make up for this, we will have a bonus, "fun" lecture sometime at the end of the semester.
  • (11/20)
    • Finished the discussion of Poincare duality (without proving everything) by saying more about compactly supported cohomology and introducing cap product. (See Section 3.3 of Hatcher for the details that I omitted.)
    • As a quick corollary, proved nondegeneracy of cup product pairing / intersection pairing on a compact oriented manifold.
    • Started proving the Lefschetz fixed point theorem for manifolds, which actually counts fixed points with signs rather than merely proving that one exists. (This will apply much of the machinery we have introduced lately. Unfortunately it's not in Hatcher.)
  • (11/25)
    • Finished the proof of the Lefschetz fixed point theorem.
    • Discussed Poincare-Lefschetz duality for a manifold with boundary.
    • As an application, proved that CP^2 is not the boundary of a compact manifold.
  • (12/2) Continuing the discussion of duality, we proved:
    • Theorem: the boundary of a 4n+1 oriented compact manifold with boundary has signature zero. (Discussed simple examples...)
    • Alexander duality for the homology of the complement of a reasonable compact subset of R^n.
  • (12/4)
    • As an application of Alexander duality, showed that a Klein bottle does not embed in R^4 (and more generally a nonorientable closed n-manifold does not embed in R^(n+1)).
    • Reviewed the course, in the framework of categories, functors, natural transformations, and homology theories.
  • (12/9) Bonus "fun" lecture. Did a bit of knot theory, namely the Kauffman bracket and the classification of rational tangles.

  Homework

  Doing exercises is essential for learning the material. Your grade will be based on homework. There will be no tests. Unfortunately, we cannot get a homework grader, so I will read your homework, but I will probably not be able to do so as thoroughly or promptly as I would like to. In any case, if you do a reasonable job on the homework, you will get a good grade. Homework will be assigned every few lectures and will be due a week or two after it is assigned.

  You can hand in homework either by bring it to class or sliding it under my office door (923 Evans). If it does not fit under my office door, please be more concise!

  • HW#1, due Friday 9/12 at 5:00 pm. Chapter 0, exercise 4. Section 1.1, exercises 6, 9, 16. Section 1.2, exercises 10, 17. Extra credit challenge problem: find an n-component link L such that L is not isotopic to an unlink, but every (n-1)-component sublink of L is isotopic to an unlink.
  • HW#2, due Friday 9/26 at 5:00 pm. Section 1.3, exercises 4, 5, 9, 12, 14. Section 1.A exercise 6.
    (I'm not assigning this, but if you're curious, Section 1.3 exercise 24 explores the analogy with Galois theory, and later exercises in Section 1.A outline more applications to group theory.)
    (Hints: for exercise 12 of Section 1.3 see Prop. 1.39, and for exercise 14 of Section 1.3 see the discussion of the infinite dihedral group on page 42.)
  • HW#3, due Friday 10/17 at 5:00 pm. Section 2.1 exercises 8, 12, 16(b), 17(b), 27, 31.
  • HW#4, due Friday 11/14 at 5:00 pm. Section 2.2 exercises 2, 7, 17. Section 2.B exercise 2. Section 2.C exercise 5. Section 3.1 exercise 9.
  • HW#5, due Friday 12/5. Section 3.2 exercise 4. Section 3.3 exercises 5, 17, 22, 32. Section 3.B exercise 4.
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