This course is a continuation of MATH 534A.
Instructor: Anton Izosimov
Classes: Tuesday and Thursday, 9:30-10:45AM in MATH 501
Office hours: Tuesday and Thursday, 11-11:50AM, and by appointment; S414, ENR2 building
Textbooks: Allen Hatcher, Algebraic Topology and John M. Lee, Introduction to Smooth Manifolds, Second Edition
Recommended problems, LaTeX source
Final exam: Tuesday, May 7, 8-10AM, in MATH 501
Tests:
Homework assignments:
- Homework 1 -- Problems 4, 7, 8, 11 from recommended problems, due on Feb 5
- Homework 2 -- Problems 16, 19, 23, 24 from recommended problems, due on Feb 19
- Homework 3 -- Problems 33, 34 from recommended problems, to be completed in groups of up to 3 people, due on Mar 19
- Homework 4 -- Problems 36, 37, 38 from recommended problems, due on Apr 2
- Homework 5 -- Problems 42, 43, 44 from recommended problems, due on Apr 30
Covered material:
- Thu, Jan 10 -- homotopy, deformation retraction, retraction, homotopy equivalence (Chapter 0 of Hatcher)
- Tue, Jan 15 -- deformation retraction as homotopy equivalence (Chapter 0 of Hatcher), paths, homotopy with fixed endpoints, concatenation of paths, definition of the fundamental group (Section 1.1 of Hatcher)
- Thu, Jan 17 -- reparametrization as homotopy, verification of group axioms for the fundamental group, dependence of the fundamental group on the basepoint (Section 1.1 of Hatcher)
- Tue, Jan 22 -- the fundamental group of the circle (Section 1.1 of Hatcher)
- Thu, Jan 24 -- proof of the homotopy lifting property (Section 1.1 of Hatcher)
- Tue, Jan 29 -- induced homomorphisms, Brouwer's fixed point theorem, homotopy invariance of the fundamental group (Section 1.1 of Hatcher)
- Thu, Jan 31 -- fundamental theorem of algebra via the fundamental group, fundamental groups of spheres (Section 1.1 of Hatcher)
- Tue, Feb 5 -- free products of groups, statement of Van Kampen's theorem, fundamental group of the figure eight (Section 1.2 of Hatcher)
- Thu, Feb 7 -- fundamental group of the torus, proof of the surjectivity part of Van Kampen's theorem (Section 1.2 of Hatcher)
- Tue, Feb 12 -- finished the proof of Van Kampen's theorem (Section 1.2 of Hatcher)
- Thu, Feb 14 -- covering spaces, homotopy lifting property, degree of the covering, fundamental group of the projective plane (Section 1.3 of Hatcher)
- Tue, Feb 19 -- universal coverings, existence and uniqueness, relation to the fundamental group (Section 1.3 of Hatcher)
- Thu, Feb 21 -- deck transformations and group actions (Section 1.3 of Hatcher)
- Thu, Feb 28 -- deck transformations and group actions (Section 1.3 of Hatcher)
- Tue, Mar 12 -- introduction to homology, simplicial homology (Section 2.1 of Hatcher)
- Thu, Mar 14 -- examples of computation of simplicial homology (Section 2.1 of Hatcher)
- Tue, Mar 19 -- definition of singular homology (Section 2.1 of Hatcher)
- Thu, Mar 21 -- homotopy invariance of singular homology (Section 2.1 of Hatcher)
- Tue, Mar 26 -- reduced homology, homology of quotients (Section 2.1 of Hatcher)
- Thu, Mar 28 -- exact sequence of a pair (Section 2.1 of Hatcher)
- Tue, Apr 2 -- the equivalence of simplicial and singular homology (Section 2.1 of Hatcher)
- Thu, Apr 4 -- Mayer-Vietoris sequence (Section 2.2 of Hatcher)
- Thu, Apr 11 -- exterior derivative (Chapter 14 of Lee)
- Tue, Apr 16 -- the Stokes theorem (Chapter 16 of Lee)
- Thu, Apr 18 -- de Rham cohomology, examples (Chapter 17 of Lee)
- Tue, Apr 23 -- de Rham cohomology of the sphere and projective plane
- Thu, Apr 25 -- homotopy invariance of de Rham cohomology (Chapter 17 of Lee), de Rham cohomology of tori
- Tue, Apr 30 -- de Rham's theorem (Chapter 18 of Lee), Poincare duality (Chapter 3.3 of Hatcher)