This course is intended for students with a strong background in point-set topology and some familiarity with the fundamental group and covering spaces. For students with weaker background, MAT 530 is more appropriate. Both courses include the same core material on the fundamental group and covering spaces, but MAT 540 discusses these concepts in the context of homotopy groups and fibrations. MAT 530 builds a solid foundation in point-set topology while MAT 540 only revisits point-set topology as necessary (for example, to establish properties of CW-spaces); students will be expected to work with point-set-topological concepts confidently and correctly. The emphasis of the course will be on the basics of homotopy theory (typically not included in introductory topology courses).

MAT 530 and MAT 540 satisfy the same requirements for the PhD and master's programs in Mathematics. MAT 540 cannot be taken for credit if you already completed MAT 530.

• Instructor: Olga Plamenevskaya, office 2-112 Math, e-mail: olga@math.stonybrook.edu

• Office hours: Monday 9:30am-10:30am, 12:30-2:30am in Math 2-112, or by appointment.

• Class meetings: MW 10:00am-11:20am, Physics P-129.

• Course webpage: http://www.math.stonybrook.edu/~olga/mat540-fall23/.

  All course information will be posted on the course webpage.

• Homework: weekly homework will be posted on this page.

  Homework 1 due Sept 13, in class.

  Homework 2 due Sept 20, in class.

  A summary of the properties of CW complexes that we discussed in class (including a lemma on continuity of homotopies).

  Homework 3 due Sept 27, in class.

  This week (9/18-9/20), we covered the cellular approximation theorem, following the Fomenko-Fuchs book. This included the free-point lemma (section 5.8) and also a similar lemma (to be used later) that allows to homotop a map between cells of equal dimension to a map that is a linear homeomorphism on the preimage of a small ball (this is the lemma in section 11.1 of Fomenko--Fuchs).

  Homework 4 due Oct 4, in class.
  This week (9/25-9/27), we discussed how attaching cells in a CW complex affects the fundamental group and the higher homotopy groups (section 11.1 Fomenko-Fuchs). In particular, we now have a strategy for computing the fundamental group of a path-connected CW complex X. Its first skeleton X^1 is homotopy equivalent to a wedge of circles (a maximal tree T in X^1 deformation retracts to a point, via a homotopy that extends to the entire X by HEP). After this homotopy, X^2 is obtained by attaching 2-cells to the wedge of circles. We have pi_1(wedge of circles) = free group on the corresponding generators, and pi_1(X^2) is obtained by quotienting this free group by the normal subgroup generated by the attaching loops of the 2-cells. Attaching 3-cells doesn't change the fundamental group, so we have pi_1(X)=pi_1(X^2). In practice, when there are just a few cells, this method works really well! This is how you compute the fundamental group of surfaces (there's only one 2-cell in the standard CW structure).

  We also covered section 5.9 in Fomenko-Fuchs: the theorem that an n-connected space is homotopy equivalent to a space without any cells in dimensions 1, 2, ..., n is really useful.

  Homework 5 due Oct 18, in class. You have two weeks - please be sure to do the reading assignments and the "do not submit" parts.
  This week (10/2-10/4), we discussed some further corollaries of the last week material: higher homotopy groups of a wedge, in a certain dimension range, for highly connected CW spaces, n-th homotopy group of a wedge on n-spheres, the strategy for computing the first non-trivial homotopy group of a CW-complex, and the statement and idea of the proof of the Hurewicz theorem (for a simply connected CW space, the first non-trivial homotopy and homology groups are computed by the same cellular procedure, so they are isomorphic; this only makes sense if you already know what homology is). This is Fomenko-Fuchs, sections 11.2, 11.3. Note that we are still accepting without proof that pi_n(S^n)=Z. We also introduced Eilenberg-Maclane spaces and proved existence (FF section 11.7).

  We also defined relative homotopy groups, discussed their properties, and proved (some parts of) homotopy exact sequence of a pair (FF sections 8.4-8.7; see also the corresponding material in Hatcher Ch 4)

  On 10/11, we proved the Whitehead theorem and discussed some corollaries (a contactible CW complex deformation retracts to a point; K(pi, n) is unique, up to homotopy). We followed Hatcher section 4.1 for Whitehead and Fomenko-Fuchs 11.7, 11.8 for Eilenberg-Maclane spaces. There was only one class that week because of the break.

  This week (10/16-10/18), we started talking about covering spaces and proved lifting theorems for paths, homotopies, and a lifting criterion for maps from a general space to the base of the covering. (See Hatcher sections 1.3 and 1.1, FF sections 6.5, 6.6.) We used coverings to compute the fundamental groups of the circle and a wedge of circles. Please read Theorems 1.8 and 1.9 in Hatcher section 1.1 if you haven't seen these applications of the fundamental group of the circle before.

  Homework 6 due Oct 25, in class.

  This week (10/23-10/25), we discussed isomorphisms of coverings, deck transformations, the correspondence between the coverings and the subgroups of the fundamental group of the base, as well as classification and hierarchy of coverings. (We will still need to prove the existence part of the classification theorem after the exam.)

  Homework 7 due Nov 1 and Nov 8, in class.

  Midterm is on Nov 1. The class on Oct 30 will be taught by Filip Samuelsen in the discussion/review format, bring your questions!

  This week (11/6-11/8), we finished the existence part of the covering spaces classification (in particular, constructed the universal covering). We started fiber bundles, discussed examples, and proved the homotopy exact sequence for Serre fibrations. Next week, we'll prove that fiber bundles have the homotopy lifting property, and then we'll know that the homotopy exact sequence actually applies in the examples we saw. This material is in Ch 4 of Hatcher, Lec 9 of Ch 1 in Fomenko-Fuchs.

  Homework 8 due Nov 15, in class.

  This week (11/13-11/15), we proved that fiber bundles have the homotopy lifting property, discussed what the homotopy sequence of a fibration says in the case of coverings. Also discussed configuration spaces of n points in R^2 and braid groups, showed that the higher homotopy groups of configuration spaces of points in R^2 are trivial, and therefore we get K(G, 1), where G is the braid group B_n (for configuration space of unordered n-tuples) or the pure braid group P_n (ordered n-tuples).

  Homework 9 due Nov 29, in class.

  This week (11/27-11/29), we proved the ("easy part of") the suspension theorem for the spheres, following Fomenko-Fuchs (with a brief discussion of the general highly connected CW space, following Hatcher). Corollaries: we finally know why the n-th homotopy group of S^n is Z. Other corollaries: the notion of degree, the Hopf theorem.

  Homework 10 due Friday, Dec 8, in class.

  During the last week (12/4-12/6), we discussed classification of surfaces. We gave two proofs, both assuming triangulation. One classical proof uses the representation of a surface as a polygon with the boundary edges identified in pairs; the proof proceeds by bringing this diagram to a standard form via a sequence of cut-and paste moves. This proof can be found in many textbooks, such as Massey's Algebraic Topology: an Introduction, Kinsey's Topology of Surfaces, etc. The second proof is shorter and more illuminating, even if it requires filling in some details (see also this Mathoverflow thread).

  The classification theorem allows to identify a compact surface easily, by checking its orientability, the number of boundary components if the surface has non-empty boundary, and the Euler characteristic (or the fundamental group, or directly the homotopy type). Many of these can be drawn in R^3; the embedding can be complicated, but the homeomorphism type can be detected easily, using the above strategy. To check for orientability, try to color the sides of the surface, paying attention to any twisted bands. You will either see that the surface is two-sided (=orientable), or if coloring two sides is impossible, then you will see that your surface contains an embedded Mobius band. (We stick to informal undertsanding of orientability at this level).
  Here are some surfaces as examples, but you can draw any number of these yourself. It's a good exercise to play with these: make deformation retraction to see the homotopy type (a surface with boundary is homotopy equivalent to a wedge of circles), compute the homotopy type and the Euler characeristic, try to see if one of the given surfaces can be a covering of another, etc.

  Going back to the classification question, tt is non-trivial to show that a triangulation exists, and/or that every topological 2-manifold is smooth. (SMOOTH and TOP are very different is dimensions greater than 3!) Here are some references for triangulation and smoothness (and a complete proof that discusses these): Hatcher's paper on the Kirby's torus trick and Tomassen's paper using Jordan Curve theorem. See also this Mathoverflow thread. We did not discuss this material in class.

  Classification of 1-manifolds: any smooth, connected 1-dimensional manifold is diffeomorphic either to the circle or to some interval of real numbers. A proof can be found in Milnor, Topology from differentiable viewpoint. The entire book is highly recommended for all the topology/geometry that it contains.

  Important: For each homework problem, please give a proof or detailed explanation as appropriate (unless otherwise stated). Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. Although you are welcome to work with others to understand how to solve the problems, you have to write all the solutions on your own, in your own words. (See Academic Integrity Statement at the bottom of the page.) You are not allowed to look for solutions online.

• Course material:
  • CW-complexes, examples and constructions of important spaces
  • Homotopy as a map, as a family of maps and as a relation. Relative homotopy. Homotopy equivalences and deformation retractions.
  • Natural group structures on sets of homotopy classes. Multiplication of paths and their homotopy classes. Homotopy properties of path multiplication.
  • Definition of fundamental group and higher homotopy groups. Homomorphisms induced by a continuous map. Homotopy groups of a product. n-connectedness. Extension of homotopy. Dependence of homotopy groups on the base point. n-simple spaces. Relative homotopy groups. Homotopy sequences of a pair and of a triple.
  • The language of bundles, fibre bundles, coverings, homotopy lifting property. Serre bundles. Homotopy sequence of a Serre bundle. Calculation of fundamental groups using covering spaces. Classification of coverings. Automorphisms of a covering.
  • CW-complexes. Cellular maps. Cellular approximation theorem. Calculation of the fundamental group of a CW-complex. Van-Kampen theorem. Homotopy classification of 1-dimensional CW-complexes. The Euler characteristic. Whitehead theorem. Homotopy excision. Suspension theorem. Simplest calculations of homotopy groups. Applications.
  • Topological manifolds without and with boundary. Interior and boundary of a manifold. Product of manifolds. Dimension. The boundary of Euclidean half-space. One- dimensional manifolds. Topological classification of one-dimensional manifolds. Triangulated two-dimensional manifolds and their topological classification.

• Textbooks:

  Fomenko, Fuchs, Homotopical Topology (GTM 273).

  This book has the advantage of discussing homotopy theory before homology (Some other standard textbooks do homology first). We will cover the material of Chapter 1 only.

  Another useful book is

  Allen Hatcher, Algebraic Topology.

  This book is available online for free download. We will cover roughly the material of Chapters 0, 1, and part of Chapter 4.

• Exams: there will be a midterm (in mid- to late October, date TBA) and a final exam.

• Grading Policy: The course grade will be determined by homeworks and class participation, one midterm, and the final exam.

  No make-up exams will be given for midterms. If a student misses a midterm exam for a well-documented medical reason or other similar circumstances beyond the student's control, the student may be excused from the exam, with the final grade determined from the other exams, homework, and class participation. For the final exam, make-ups will be given ONLY in cases of properly documented medical reasons or other similar circumstances, at the instructor's discretion.

  
  Academic Integrity Statement Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html.

  
  Critical Incident Management Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Student Conduct and Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures. Further information about most academic matters can be found in the Undergraduate Bulletin, the Undergraduate Class Schedule, and the Faculty-Employee Handbook.

  
  Student Accessibility Support Center Statement If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.