## Math 6620 -- Algebraic Topology -- W. Cherry

Class Meets:   MWF 11-11:50 in GAB 461.
Final Exam:   10:30-12:30 Monday, December 11 in GAB 461.

## Instructor: William Cherry

 Office Location: General Academic Building (GAB) 405. Office Hours: Mondays 12:30-1:30 Tuesdays & Thursdays 1:00-3:00 Wednesdays 9:30-10:30, 12:30-1:30, and 4-5 and by appointment.

## Textbook

Glen Bredon, Topology and Geometry, Springer, 1993.

## Trouble reading or printing course hand-outs

Many of the links below are to course hand-outs in Adobe PDF (or Acrobat) format. If you are having trouble reading or printing these hand-outs, click here.

Course syllabus

## Optional Homework

 Date Assigned Due not later than Assignment details Aug 28 Sep 11 I.2#2, I.3#1 and 6, I.4, #1 and 5. Also if A is a subset of a topological space X, is the subspace toplogy on A the largest topology on A such that the inclusion map is continuous? The smallest? Neither one? Sep 6 Sep 22 I.13#3-7. Also, if you have some familiarity with complex numbers, check that CP1 is homeomorphic to S2. Sep 11 Sep 22 I.14#1, 3, 4, 6, 9. Sep 20 Sep 29 Gluing and III.2.3 Sep 27 Oct 4 III.3: 2-4 Oct 11 Oct 20 III.9: 1, 3-8, 10-13 (these last few are not so easy) Oct 25 Nov 1 IV.6: 2-5 Oct 27 Nov 10 Prove that the cylinder S1×[0,1] is not homeomorphic to the Möbius band.

## Lecture Summaries

 Date Topic Review of and Crash Course in General Topology Aug 28 Topological Vocabulary: Ch. I, Sections 2-3 Aug 30 Mostly connectedness: Ch. I, Section 4; plus definition of Hausdorff: Ch. I, Section 5 Sep 1 Compactness, proper mappings, product topology: Ch. I, Sections 7-8 Sep 6 Quotient topology: Ch. I, Section 13 Sep 8 Homotopy: Ch. I, Section 14 Sep 11 More Homotopy: Ch. I, Section 14 A brief tour of differentiable manifolds: This material provides motivation and examples for what we will be studying. For those interested, an undergraduate level treatment can be found in Guillemin and Pollack, Differential Topology, which is inspired by the well-known introduction by Milnor, Topology from the Differentiable Viewpoint. Sep 13 Definitions of (smooth) manifold and examples: Ch. II, Sections 2-4 Sep 15 Poincare Conjecture, Exotic Spheres, the graph of the absolute value function in R2 as a manifold (Exercise II.2.4: compare with the discussion of the cone following Definition II.4.5), manifold with boundary (Definition II.2.7), brief discussion without proof of classification of one and two dimensional compact connected manifolds, allowing boundary in the 1-dimensional case, tangent vectors & differentials (Ch. II, Sec. 5), Tangent Bundles (Ch. II, Sec. 9), and the Whitney Embedding Theorem without proof (Ch. II, Sec. 10) Homotopy & Fundamental Group Sep 18 & 20 Definition of fundamental group and the computation of the fundamental group for R2\{(0,0)} using "net change in argument" (or winding numbers). This is a variation on Ch. III, Sec. 1-2. Sep 22 Spheres are simply connected (Th. III.2.2) and fundamental group of a product space (Th. III.2.6) Sep 25 Covering Spaces: Ch. III, Sec. 3 Sep 27 The Lifting Theorem: Ch. III, Secs. 4 & 5 Sep 29 Deck Transformations: Ch. III, Sec. 6 Oct 2 Seifert-Van Kampen: Ch. III, Sec. 9 Oct 4 Properly discontinuous actions (Ch. III, Sec. 7), Lens Spaces, Classification & existence of coverings (Ch. III, Sec. 8), SO(3,R) (Ch. III, Sec. 10) Oct 6 TEST: The test will emphasize definitions, vocabulary, statements of theorems, and "easy" consequences, for example calculating the fundamental group of RP2. It will cover Ch. I, Sec. 14 (homotopy), and Ch. III, Secs. 1-6 (fundamental group & covering spaces). Click here for the test solutions. Singular Homology Oct 9 Singular simplices & definition of homology: Ch. IV, Sec. 1 Oct 11 Augmentation map, zero homology group, and reduced homology: Ch. IV, Sec. 2 Oct 13 H1 is isomorphic to the Abelianized fundamental group: Ch. IV, Sec. 3 Oct 16 A less than successful introduction to homological algebra and diagram chasing: Ch. IV, Sec. 4 Oct 18 A more successful day diagram chasing: Ch. IV, Sec. 4 Oct 20 Eilenberg-Steenrod-Milnor Axioms: Ch. IV, Sec. 6 Oct 23 Homology of Spheres & Brower Fixed Point Theorem: Ch. IV, Sec. 6 Oct 25 Degree of the antipodal map & consequences for even dimensional spheres: Ch. IV, Sec. 6 Oct 27 Local Homology Hi(X,X\{x}); Invariance of dimension (see the discussion following Corollary IV.19.10); Homology of RP2: Ch. IV, Sec. 7 Oct 30 CW Complexes (definition) and examples: Ch. IV, Sec. 8 Nov 1 Start toward cellular homology: Lemma IV.10.1 Nov 3 TEST: Chapter IV, Sections 1-6 (up to and including 6.14), plus Section 8. Click here for the test solutions. Nov 6 Cellular homology: page 202 and a sketch of Theorem 10.3. Homology of real projective space: Ch. IV, Sec. 14 Nov 8 Chain homotopy and singular homology of contractible spaces: Ch. IV, Sec. 15 Nov 10 Cross product and the proof that singular homology satisfies the homotopy axiom: Ch. IV, Sec. 16 Nov 13 Subdivision, Excision, and Mayer-Vietoris: Ch. IV, Sec. 17-18 Nov 15 Embeddings of disks and spheres into spheres: Ch. IV, Sec 19; consequences of the Borsuk-Ulam Theorem: Ch. IV, Sec 20 Nov 17 Borsuk-Ulam Theorem: Ch. IV, Sec 20 Cohomology Nov 20 Differential forms & de Rham cohomology: Ch. V Nov 22 Differential forms & de Rham cohomology: Ch. V Nov 27 de Rham's Theorem: Ch. V, Sec. 9 Nov 29 Universal Coefficient Theorem: Ch. V, Sec. 7 Dec 1 TEST: Know the definitions of the following concepts and statements of the following theorems: The cross product theorem (Th. IV.16.1), Mayer-Vietoris (Th. IV.18.1), Generalized Jordan Curve Theorem (Th. IV.19.4), Invariance of Domain (Cor. IV.19.9), Borsuk-Ulam Theorem and its consequences (Ch. IV, Sec. 20), differential form, exterior derivative, de Rham cohomology, singular cohomology, de Rham's Theorem (Th. V.9.1) Dec 5 Cross, Kroneker, cup, and cap products. Kunneth formula. Ch. VI Dec 7 Compactly supported de Rham cohomology, Poincare duality from the de Rham perspective, intersection theory Dec 11 FINAL (open book): Expect exercises from the following sections of the book: I.14, III.2-6, III.9, IV.4, IV.10, IV.18, IV.20. Expect some of these problems verbatim and some using similar techniques.