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.


Glen Bredon, Topology and Geometry, Springer, 1993.

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Course Hand-Outs

Course syllabus

Optional Homework

Date AssignedDue not later than Assignment details
Aug 28Sep 11I.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 6Sep 22I.13#3-7. Also, if you have some familiarity with complex numbers, check that CP1 is homeomorphic to S2.
Sep 11Sep 22I.14#1, 3, 4, 6, 9.
Sep 20Sep 29 Gluing and III.2.3
Sep 27Oct 4 III.3: 2-4
Oct 11Oct 20 III.9: 1, 3-8, 10-13 (these last few are not so easy)
Oct 25Nov 1 IV.6: 2-5
Oct 27Nov 10 Prove that the cylinder S1×[0,1] is not homeomorphic to the Möbius band.

Lecture Summaries

Review of and Crash Course in General Topology
Aug 28Topological Vocabulary: Ch. I, Sections 2-3
Aug 30Mostly connectedness: Ch. I, Section 4; plus definition of Hausdorff: Ch. I, Section 5
Sep 1Compactness, proper mappings, product topology: Ch. I, Sections 7-8
Sep 6Quotient topology: Ch. I, Section 13
Sep 8Homotopy: Ch. I, Section 14
Sep 11More 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 13Definitions of (smooth) manifold and examples: Ch. II, Sections 2-4
Sep 15Poincare 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 & 20Definition 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 22Spheres are simply connected (Th. III.2.2) and fundamental group of a product space (Th. III.2.6)
Sep 25Covering Spaces: Ch. III, Sec. 3
Sep 27The Lifting Theorem: Ch. III, Secs. 4 & 5
Sep 29Deck Transformations: Ch. III, Sec. 6
Oct 2Seifert-Van Kampen: Ch. III, Sec. 9
Oct 4Properly discontinuous actions (Ch. III, Sec. 7), Lens Spaces, Classification & existence of coverings (Ch. III, Sec. 8), SO(3,R) (Ch. III, Sec. 10)
Oct 6TEST: 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 9Singular simplices & definition of homology: Ch. IV, Sec. 1
Oct 11Augmentation map, zero homology group, and reduced homology: Ch. IV, Sec. 2
Oct 13H1 is isomorphic to the Abelianized fundamental group: Ch. IV, Sec. 3
Oct 16A less than successful introduction to homological algebra and diagram chasing: Ch. IV, Sec. 4
Oct 18A more successful day diagram chasing: Ch. IV, Sec. 4
Oct 20Eilenberg-Steenrod-Milnor Axioms: Ch. IV, Sec. 6
Oct 23Homology of Spheres & Brower Fixed Point Theorem: Ch. IV, Sec. 6
Oct 25Degree of the antipodal map & consequences for even dimensional spheres: Ch. IV, Sec. 6
Oct 27Local Homology Hi(X,X\{x}); Invariance of dimension (see the discussion following Corollary IV.19.10); Homology of RP2: Ch. IV, Sec. 7
Oct 30CW Complexes (definition) and examples: Ch. IV, Sec. 8
Nov 1Start toward cellular homology: Lemma IV.10.1
Nov 3TEST: Chapter IV, Sections 1-6 (up to and including 6.14), plus Section 8. Click here for the test solutions.
Nov 6Cellular homology: page 202 and a sketch of Theorem 10.3. Homology of real projective space: Ch. IV, Sec. 14
Nov 8Chain homotopy and singular homology of contractible spaces: Ch. IV, Sec. 15
Nov 10Cross product and the proof that singular homology satisfies the homotopy axiom: Ch. IV, Sec. 16
Nov 13Subdivision, Excision, and Mayer-Vietoris: Ch. IV, Sec. 17-18
Nov 15Embeddings of disks and spheres into spheres: Ch. IV, Sec 19; consequences of the Borsuk-Ulam Theorem: Ch. IV, Sec 20
Nov 17Borsuk-Ulam Theorem: Ch. IV, Sec 20
Nov 20Differential forms & de Rham cohomology: Ch. V
Nov 22Differential forms & de Rham cohomology: Ch. V
Nov 27de Rham's Theorem: Ch. V, Sec. 9
Nov 29Universal Coefficient Theorem: Ch. V, Sec. 7
Dec 1TEST: 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 5Cross, Kroneker, cup, and cap products. Kunneth formula. Ch. VI
Dec 7Compactly supported de Rham cohomology, Poincare duality from the de Rham perspective, intersection theory
Dec 11FINAL (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.

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