# Math 215a: Algebraic topology

## Instructor

Michael Hutchings
Email: [my last name minus the last letter]@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Tu 2-5.

## Spacetime coordinates

TuTh, 9:30-11:00. As of 9/8, we are officially moving to 9 Evans.

## Prerequisites

The only formal requirements are some basic algebra, point-set topology, and "mathematical maturity". However, the more familiarity you have with algebra and topology, the better.

## Syllabus

Algebraic topology seeks to capture the "essence" of a topological space in terms of various algebraic and combinatorial objects. We will construct three such gadgets: the fundamental group, homology groups, and the cohomology ring. We will apply these to prove various classical results such as the classification of surfaces, the Brouwer fixed point theorem, the Jordan curve theorem, the Lefschetz fixed point theorem, and more.

An important topic related to algebraic topology is differential topology, i.e. the study of smooth manifolds. In fact, I don't think it really makes sense to study one without the other. So without making differential topology a prerequisite, I will emphasize the topology of manifolds, in order to provide more intuition and applications.

No single textbook does all the things that I want to do in this course. The official textbook is Algebraic Topology by Hatcher. This is a very nice book, although it does not say much about differential topology. Another very nice algebraic topology text which also covers some differential topology is Geometry and Topology by Bredon. Some further recommended books are listed below.

## Homework

Doing homework exercises is essential for learning the material, and is an official course requirement. A reasonable effort on homework will result in a good grade.
• HW#1, due Thursday 9/15 at the beginning of class. pdf ps
• HW#1a, due Thursday 9/22 at the beginning of class: Grade HW#1 according to the guidelines below. Here are solutions.
• HW#2, due Thursday 9/29 at the beginning of class. pdf ps
• HW#3, due Thursday 10/13 at the beginning of class. pdf
• HW#3a, due Thursday 10/20 at the beginning of class: Grade HW#3. Here are solutions.
• HW#4, due Thursday 10/27 at the beginning of class. pdf
• HW#4a, due Thursday 11/3 at the beginning of class: Grade HW#4. Here are solutions.
• HW#5, due Tuesday 11/15 at the beginning of class. pdf
• HW#5a, due Tuesday 11/22 at the beginning of class: Grade HW#5. Here are solutions.
• HW#6, due Tuesday 11/29 at the beginning of class. pdf (Note: some of the problems use the notion of "fundamental class", which we will discuss in class on 11/22, and which is explained on p. 236 of Hatcher.)
• HW#6a, due Tuesday 11/36 (that's a joke, please don't write to me and complain) at the beginning of class: Grade HW#6. Here are solutions.
• HW#7, due never (because there won't be time to grade it): select some interesting problems from section 3.3 of Hatcher and do them.
Homework grading policy. Since there are approximately fifty students in the class, it is not possible for me to read all of the homework carefully, and unfortunately I was not able to get a grader. So we will try the following, which might even be more educational than the traditional approach. Each homework assignment will be due at the beginning of class on some particular day. At the end of that lecture, everyone who handed in an assignment will be given a random assignment from someone else to grade. (I will keep a record of who gets which assignment.) After class, I will post solutions online to help with grading (although of course these solutions are not unique). If you can't make it to class on the day that the homework is due, please slide your homework under my office door before class, together with a note explaining how I can give you another assignment to grade.

To grade the assignment, please take a separate sheet of paper, and label it with both your name and the name of the person whose assignment you are grading. Write at most one page of commentary on the assignment. If an answer is good, say so; if you are not convinced by a proof, say why. Then give the whole assignment a grade of 3 (excellent), 2 (good), or 1 (needs improvement). The grades should not be taken too seriously; the important thing is to communicate and critique mathematical arguments. (At the end of the course, I will play some statistical games to adjust for the fact that different people grade more generously than others.)

Finally, staple your commentary to the assignment, and hand in the package to me in or before class one week after the assignment was due. I will then look over all of the graded assignments and return them in a subsequent class.

## What I plan to do

• Introductory lecture.

• The fundamental group and covering spaces. (about 7 lectures)

• The fundamental group. (See Hatcher, sections 1.1 and 1.2.)
• Definition and basic examples: R^n, S^1 (by lifting to R), products, 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).
• A more interesting example: the braid group. (Not in Hatcher.)
• Applications of the fundamental group of the circle:
• The Brouwer fixed point theorem and Borsuk-Ulam theorem in two dimensions.
• The so-called fundamental theorem of algebra.
• The Seifert-van Kampen theorem for the fundamental group of the union of two spaces.
• 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. (For details about the topology of CW complexes, see the Appendix in Hatcher.)
• Dependence of the fundamental group on the base point.
• A homotopy equivalence induces an isomorphism on fudamental groups.
• The trefoil knot is knotted and the Hopf link and Borromean rings are linked. (Not in Hatcher.)

• Covering spaces. (See Hatcher, section 1.3.)
• Definition and discussion of basic examples, including the relation between finite covers of S^1 and permutations, coverings of the torus, the covering of any genus g>1 surface by the hyperbolic plane, the canonical double covering of any manifold by an oriented manifold, and the non-example of branched covers of Riemann surfaces.
• Lifting paths to a covering space.
• The lifting criterion for maps to a covering space.
• The classification of covering spaces over a reasonable space.
• An application to algebra: any subgroup of a free group is free.

• Brief introduction to higher homotopy groups: definition, easy properties, introduction to pi_n(S^n) (which will be discussed more later), why homotopy groups are hard in general.

• Homology. (about 13 lectures; see Hatcher, chapter 2)

• Introducing singular homology.
• Warmup definition: simplicial homology of a Delta-complex.
• Main definition: singular homology of a topological space.
• H_0 is a direct sum of Z's, one for each path component.
• Computation of the homology of a contractible space, using cones over simplices.
• H_1 of a path connected space is the abelianization of pi_1. (I prove this differently from Hatcher, without using the classificaion of surfaces, with generalization to the Hurewicz theorem in mind.)
• Homology is a homotopy invariant. (I prove this a bit differently from Hatcher, using "acyclic models" to avoid explicitly constructing a "triangulation" of a prism, by using homology theory to show that a "triangulation" with the desired properties exists. One can also avoid this issue by defining singular homology using cubes instead of simplices.)
• Mayer-Vietoris sequence:
• statement and simple examples, including the homology of S^n.
• A bit of homological algebra: a short exact sequence of chain complexes induces a long exact sequence in homology.
• Proof of Mayer-Vietoris sequence using subdivision lemma.
• Understanding the connecting homomorphism in the Mayer-Vietoris sequence.
• Application: proof of the generalized Jordan curve theorem.
• An interesting example: linking numbers.
• How to compute the homology of a CW complex.
• Relative homology; long exact sequences of a pair and a triple; excision; isomorphism between relative homology and reduced homology of the quotient for "good pairs".
• Equivalence of simplicial and singular homology.
• Cellular homology of a CW complex.
• Extensive discussion (including some perspectives not in Hatcher) of the degree of a map from a sphere to itself.
• Application to the "hairy ball theorem".
• Isomorphism between singular and cellular homology of a CW complex.
• Examples including real and complex projective space.
• Euler characteristic, and Lefschetz fixed point theorem for simplicial complexes.
• Homology with coefficients, Tor, and the universal coefficient theorem.
• The Kuenneth formula for the homology of the product of two spaces, via the Eilenberg-Zilber theorem. (The approach I am using here is different from that of Hatcher, and may be found in Bredon.)

• Cohomology. (about 7 lectures; see Hatcher, chapter 3)

• Definition of cohomology, and universal coefficient theorem for cohomology.
• Definition of the cup product.
• Statement of Poincare duality (to be proved shortly).
• Statement of the geometric interpretation of cup product via intersection theory in a smooth manifold. (The proof is beyond the scope of this course, but I believe that the statement is the most important thing to know about cup product.)
• Computation of the cohomology ring of projective spaces.
• Application: the Borsuk-Ulam theorem and the Ham Sandwich Theorem.
• Main ideas in the proof of Poincare duality, including the orientation bundle, compactly supported cohomology, direct limits, and the cap product.
• Application: the Lefschetz fixed point theorem on a smooth manifold (which actually counts fixed points with signs, rather than merely proving that a fixed point exists). (This is not in Hatcher, but it is in Bott-Tu or Bredon.)
• Poincare-Lefschetz duality for a manifold with boundary.
• Application: the boundary of a 4n+1 dimensional oriented compact manifold has signature zero.
• Alexander duality.
• Application: the Klein bottle does not embed in R^3 (and more generally, a nonorientable closed n-manifold does not embed in R^(n+1)).

• Concluding lecture.

## What we actually did.

• (8/30) Introduced some of the problems that algebraic topology can solve, or try to solve. Gave an overview of the classification of manifolds. Algebraic topology can also be used to prove existence theorems, such as the Brouwer fixed point theorem and the Borsuk-Ulam theorem. It can also be used to count things, as in the Lefschetz fixed point theorem.

• (9/1) Defined the fundamental group. Computed the fundamental group of the circle, by lifting paths to the real line. As applications, outlined proofs of the Brouwer fixed point theorem in two dimensions, the Borsuk-Ulam theorem in two dimensions, and the "fundamental theorem of algebra".

• (9/6) Discussed the extent to which the fundamental group depends on the base point, and proved that the fundamental group is invariant under homotopy equivalence. Computed the fundamental group of a product, and of S^n for n>1 (in two different ways).

• (9/8) Explained the Seifert-van Kampen theorem for the fundamental group of a union. (I didn't prove the whole thing; see Hatcher and the other references for the complete proof.) Computed examples, especially surfaces. The punchline, which will be explained more next time, is that if X is a CW complex with one 0-cell x_0, then pi_1(X,x_0) has a natural presentation with one generator for each 1-cell and one relation for each 2-cell.

• (9/13) Explained the punchline from the previous lecture. (See Hatcher for more details.) Had some fun with the Wirtinger presentation of the fundamental group of a knot complement, and the braid group. (These topics are not discussed much in Hatcher, and we will not use them much later.)

• (9/15) Introduced covering spaces and examples. Discussed the path lifting lemma and homotopy lifting lemma. Showed that a covering space determines both a right action of the fundamental group of the base on the fiber over the base point, and (given a choice of a base point upstairs in the fiber over the base point downstairs) a subgroup of the fundamental group of the base.

• (9/20) Studied examples of covering spaces (of the torus and the figure eight) and their associated subgroups of the fundamental group of the base and right actions of the fundamental group of the base on the fiber. Explained the Lifting Criterion.

• (9/22) Explained the classification of path connected covering spaces of a reasonable space in terms of subgroups of the fundamental group. (There is another classification of covering spaces in terms of right actions of the fundamental group, which I didn't have time to explain.) As an application, proved that any subgroup of a free group is free.

• (9/27) Introduced the simplicial homology of a Delta-complex. (From now on, the algebra part of the course will be linear algebra instead of group theory, which is a lot easier, although not as easy as you might think.)

• (9/29) Defined the singular homology of a topological space. Showed that H_0 of a (nonempty) path connected space is Z. Computed the homology of a contractible space. (I departed from the book here.) Started to explain why H_1 of a path connected space is the abelianization of pi_1.

• (10/4) Completed the proof that H_1 is the abelianization of pi_1. Proved homotopy invariance of homology. (I did this differently from the book, without having to explicitly construct the prism operator; the argument I gave is in Greenberg and Harper, and in greater generality in Bredon.)

• (10/6) Introduced the Mayer-Vietoris sequence, used it to compute some examples of homology, and discussed how a short exact sequence of chain complexes induces a long exact sequence in homology.

• (10/11) Proved (most of) the Subdivision Lemma. Used it to derive the Mayer-Vietoris sequence.

• (10/13) Explained the Jordan-Brouwer separation theorem. Elaborated on this by discussing the linking number of two smooth knots.

• (10/18) Introduced relative homology. Excision and applications. Isomorphism between simplicial and singular homology of a Delta-complex, using the Five Lemma.

• (10/20) Discussed the degree of a map from a sphere to itself, from various points of view. Introduced the cellular homology of a CW complex.

• (10/25) Discussed the cellular homology of a CW complex a lot more. Introduced the Euler characteristic of a finite CW complex.

• (10/27) Proof that the Euler characteristic is a topological invariant. Generalization to the Lefschetz number of a map. Statement and sketch of proof of Lefschetz fixed point theorem for a finite simplicial complex.

• (11/1) Homology with coefficients, the universal coefficient theorem, and Tor. Cf. Hatcher, section 3.A. If you are curious, you can read about the more general theory of derived functors in Lang's Algebra.

• (11/3) Explained the Kuenneth formula for the homology of a Cartesian product, following the approach in Bredon. It is instructive to compare this with the proof for CW complexes in Hatcher, section 3.B.

• (11/8) Introduced cohomology. Proved the universal coefficient theorem for cohomology.

• (11/10) Introduced the cup product on cohomology.

• (11/15) Used the cup product to prove the Borsuk-Ulam theorem. Discussed the Kuenneth formula for cohomology.

• (11/17) Explained the product structure on the cohomology of a Cartesian product. Introduced orientations of manifolds.

• (11/22) More about orientation. Explained the fundamental class of an R-oriented compact manifold. More generally showed that if A is a compact subset of an n-dimensional manifold M, then H_n(M,M-A;R) is canonically isomorphic to the sections of the R-orientation bundle over A. (This is in Bredon.) Gave more details about direct limits.

• (11/29) Discussed the main ideas in the proof of Poincare duality, including the cap product and compactly supported cohomology.

• (12/1) Brief review of smooth manifolds. Stated duality between cup product and intersection of submanifolds. (See sections 1 and 2 of this handout. The rest of the handout uses material which you can learn about in 215b.)

• (12/6) Proved the Lefschetz fixed point theorem on a smooth manifold (cf. Bredon or above handout). Corollary: Poincare-Hopf index theorem.

• (12/8) Introduced Poincare-Lefschetz duality for a manifold with boundary. Used this to show that the boundary of a compact oriented (4k+1)-dimensional manifold has signature zero. Didn't have time to explain Alexander duality, but you should read about this in section 3.3 of Hatcher.

## References

The following are some related books which are sitting on my shelf, along with my opinions about them. Some of these may be useful now, while others will be more useful after you have taken 215a.

• C. Adams, The knot book. Every topologist should know a little knot theory, if for no other reason than to be able to explain something about what you do to non-mathematicians. In the course I will do a couple of examples of simple applications of algebraic topology to knot theory. This book is an easy read. Why study hard algebraic topology, if you haven't learned this easier stuff first?
• Bott and Tu, Differential forms in algebraic topology. This book gives a beautiful tour through many topics including de Rham theory, Cech cohomology, basic homotopy theory, spectral sequences, Chern classes. The path taken by this book is very different from most other algebraic topology textbooks. The basic concepts are introduced using manifolds and differential forms; this sacrifices some generality, but is more relevant for many geometric applications, and allows for much more intuitive explanations.
• G. Bredon, Geometry and Topology. This is a nice algebraic topology text with a welcome emphasis on manifolds. It provides a nice alternate perspective on the basic material covered in Hatcher.
• J. Dieudonne, A history of algebraic and differential topology, 1900-1960. This book is very interesting if you are already familiar with the basic concepts of algebraic topology and want to learn how people originally thought of them. Don't be put off by the word "history" in the title; it gives excellent summary explanations of many mathematical topics.
• Gompf and Stipcisz, 4-manifolds and Kirby calculus. This book leads up to some current research topics in 4-dimensional topology, and has lots of pictures. You should be able to understand this after taking 215.
• Greenberg and Harper, Algebraic topology: a first course . This was the primary textbook when I took algebraic topology. It provides a nice concise development of singular homology theory.
• Guillemin and Pollack, Differential topology. This is a gorgeous book on basic differential topology. I highly recommend reading this, and the prerequisites are minimal.
• A. Hatcher, Algebraic topology. This is a very nice text. I wish it would do a bit more with manifolds, because I think that this adds a lot of insight. But, no single book can do everything.
• W.S. Massey has written at least three algebraic topology texts at different levels, of which I have seen and enjoyed two. If I recall correctly, he gives nice treatments of 2-manifolds, the fundamental group, and cubical singular homology.
• J.P. May, A concise course in algebraic topology. Provides a nice summary of algebraic topology from a certain advanced viewpoint. (For example, some pretty heavy category theory appears before the fundamental group of a surface is computed.)
• J. McCleary, A user's guide to spectral sequences . Too advanced for 215a, but might be good for 215b. A very useful book about spectral sequences.
• Milnor and Stasheff, Characteristic classes. A beautiful treatment of vector bundles and characteristic classes. We won't get to this material in 215a, but I have taught it in 215b.
• J. Munkres, Topology (2nd edition) . This book gives a clear and gentle treatement which should be good for beginners. The first part of the book covers point-set topology. We will be assuming basic point-set topology in the course (although this book does more of it than we will need). The second part of the book introduces the beginnings of algebraic topology.
• D. Rolfsen, Knots and links. After reading the Adams book, if you want to see some more serious applications of algebraic topology to knot theory, this book is a classic.
• E. Spanier, Algebraic topology. Kind of heavy going, but a good reference.
• M. Spivak, A comprehensive introduction to differential geometry, volume I. This book gives an obsessively detailed explanation of the basic concepts of differential topology. The other four volumes in the series (which are not really about topology) are also an interesting read. The way I read them was to start with the last chapter of volume 5 and look things up in earlier chapters as needed.

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