Math 490 Lecture Log and Homework Assignments

Math 490: Lecture Log and Homework Assignments

All references are to the course textbook (Kinsey), by chapter and section, unless otherwise specified.

Date Topics covered Reference Remarks

8/22 Introduction and overview, fundamental problem of topology, branches of topology, Euler characteristic Ch 1 Welcome to Math 490!

8/24 Euler characteristic, Euler's formula, regular polyhedra Ch 1 8/24/07 handout on polyhedra

8/27 Point-set topology: open and closed sets, relatively open and closed sets, topological spaces Ch 2.1, 2.2, 3.1

8/29 Point-set topology: continuity, homeomorphisms Ch 2.3, 3.2

8/31 Point-set topology: homeomorphisms (continued), quotient topology Ch 3.5

9/5 Point-set topology: connectedness, applications Ch 2.5, 3.2 Homework #1 due

9/7 Point-set topology: compactness Ch 2.4, 3.2

9/10 Point-set topology: compactness Ch 2.4, 3.2

9/12 Review of point-set topology Ch 2,3 Homework #2 due

9/14 Point-set topology: separation axioms, normality, partitions of unity Ch 3.3

9/17 Manifolds and surfaces, embedding theorem Ch 4.3

9/19 Planar diagrams, cellular 2-complexes, connected sum Ch 4.1, 4.2, 4.3

9/21 Connected sum, algebra of connected sum operation, examples (torus, projective plane, Klein bottle) Ch 4.3, 4.5

9/24 Examples of connected sum, standard models for surfaces, Classification Theorem for compact, connected surfaces without boundary Ch 4.3, 4.5 Homework #3 due

9/26 Proof of the classification theorem I Ch 4.5

9/28 Proof of the classification theorem II none Guest lecturer: Prof. N. Dunfield

10/1 Proof of the classification theorem III, Euler characteristic Ch 5.3, 5.4

10/3 Simplicial Gauss-Bonnet theorem [Bloch, Ch. 3.7] Homework #4 due

10/5 Map coloring and graph embedding on surfaces Ch 5.5, 5.6

10/8 Map coloring and graph embedding on surfaces II Ch 5.5, 5.6

10/10 Homology: chain groups, boundary operator Ch 6.1

10/12 Homology: cycle groups, boundary groups, homology groups, examples Ch 6.3 Homework #5 due

10/15 Homology: more examples, orientability and H_2 Ch 6.4

10/17 Betti numbers, Euler-Poincare formula, cellular functions and their induced action on homology groups Ch 6.5, 7.1 Midterm Exam on Thursday 10/18, 5:00 - 6:30 pm

10/19 Cellular functions and homology: examples Ch 7.3

10/22 Covering spaces Ch 7.4 Notices of the AMS article about Boy's surface

10/24 Affine Euclidean geometry, convex hull, definition of a simplex, symmetry groups and orientation, action of the boundary operator on chain groups and induced chain maps arising from a cellular function Ch 6.2,6.3,7.1,7.2

10/26 NO CLASS

10/29 Invariance of homology, ordinary differential equations and vector fields in R^2 Ch 8.1, 11.1

10/31 ODE and vector fields, winding number, index of a critical point Ch 11.1

11/2 Brouwer Fixed Point Theorem, Poincare-Bendixson Theorem, Poincare Index Theorem (for the disc), applications in complex analysis Ch 11.1

11/5 Introduction to differentiable manifolds, tangent space, vector fields on a manifold, Ch 11.2, 11.3

11/7 Vector fields on surfaces, Poincare Index theorem, gradient vector fields and Morse theory Ch 11.3 An article in the MIT Undergraduate Math Journal on Euler characteristic, classification of surfaces, Poincare Index Theorem, etc.

11/9 Vector fields on surfaces (continued) Ch 11.3 Homework #6 due

11/12 Hopf trace theorem and Lefschetz fixed point theorem: statements and applications [Henle, section 36]

11/14 Hopf trace theorem and Lefschetz fixed point theorem: proofs [Henle, section 36]

11/16 Introduction to knot theory [Adams, Ch 1] Homework #7 due

11/26 Elementary knot invariants [Adams, Ch 1]

11/28 Seifert fibered surfaces [Adams, Ch 4]

11/30 genus of a knot, torus knots, 3-manifolds, lens spaces, homology spheres, Poincare conjecture, Geometrization conjecture, Thurston geometries [Adams, Ch 5,9]

Date	Topics covered	Reference	Remarks
8/22	Introduction and overview, fundamental problem of topology, branches of topology, Euler characteristic	Ch 1	Welcome to Math 490!
8/24	Euler characteristic, Euler's formula, regular polyhedra	Ch 1	8/24/07 handout on polyhedra
8/27	Point-set topology: open and closed sets, relatively open and closed sets, topological spaces	Ch 2.1, 2.2, 3.1
8/29	Point-set topology: continuity, homeomorphisms	Ch 2.3, 3.2
8/31	Point-set topology: homeomorphisms (continued), quotient topology	Ch 3.5
9/5	Point-set topology: connectedness, applications	Ch 2.5, 3.2	Homework #1 due
9/7	Point-set topology: compactness	Ch 2.4, 3.2
9/10	Point-set topology: compactness	Ch 2.4, 3.2
9/12	Review of point-set topology	Ch 2,3	Homework #2 due
9/14	Point-set topology: separation axioms, normality, partitions of unity	Ch 3.3
9/17	Manifolds and surfaces, embedding theorem	Ch 4.3
9/19	Planar diagrams, cellular 2-complexes, connected sum	Ch 4.1, 4.2, 4.3
9/21	Connected sum, algebra of connected sum operation, examples (torus, projective plane, Klein bottle)	Ch 4.3, 4.5
9/24	Examples of connected sum, standard models for surfaces, Classification Theorem for compact, connected surfaces without boundary	Ch 4.3, 4.5	Homework #3 due
9/26	Proof of the classification theorem I	Ch 4.5
9/28	Proof of the classification theorem II	none	Guest lecturer: Prof. N. Dunfield
10/1	Proof of the classification theorem III, Euler characteristic	Ch 5.3, 5.4
10/3	Simplicial Gauss-Bonnet theorem	[Bloch, Ch. 3.7]	Homework #4 due
10/5	Map coloring and graph embedding on surfaces	Ch 5.5, 5.6
10/8	Map coloring and graph embedding on surfaces II	Ch 5.5, 5.6
10/10	Homology: chain groups, boundary operator	Ch 6.1
10/12	Homology: cycle groups, boundary groups, homology groups, examples	Ch 6.3	Homework #5 due
10/15	Homology: more examples, orientability and H_2	Ch 6.4
10/17	Betti numbers, Euler-Poincare formula, cellular functions and their induced action on homology groups	Ch 6.5, 7.1	Midterm Exam on Thursday 10/18, 5:00 - 6:30 pm
10/19	Cellular functions and homology: examples	Ch 7.3
10/22	Covering spaces	Ch 7.4	Notices of the AMS article about Boy's surface
10/24	Affine Euclidean geometry, convex hull, definition of a simplex, symmetry groups and orientation, action of the boundary operator on chain groups and induced chain maps arising from a cellular function	Ch 6.2,6.3,7.1,7.2
10/26	NO CLASS
10/29	Invariance of homology, ordinary differential equations and vector fields in R^2	Ch 8.1, 11.1
10/31	ODE and vector fields, winding number, index of a critical point	Ch 11.1
11/2	Brouwer Fixed Point Theorem, Poincare-Bendixson Theorem, Poincare Index Theorem (for the disc), applications in complex analysis	Ch 11.1
11/5	Introduction to differentiable manifolds, tangent space, vector fields on a manifold,	Ch 11.2, 11.3
11/7	Vector fields on surfaces, Poincare Index theorem, gradient vector fields and Morse theory	Ch 11.3	An article in the MIT Undergraduate Math Journal on Euler characteristic, classification of surfaces, Poincare Index Theorem, etc.
11/9	Vector fields on surfaces (continued)	Ch 11.3	Homework #6 due
11/12	Hopf trace theorem and Lefschetz fixed point theorem: statements and applications	[Henle, section 36]
11/14	Hopf trace theorem and Lefschetz fixed point theorem: proofs	[Henle, section 36]
11/16	Introduction to knot theory	[Adams, Ch 1]	Homework #7 due
11/26	Elementary knot invariants	[Adams, Ch 1]
11/28	Seifert fibered surfaces	[Adams, Ch 4]
11/30	genus of a knot, torus knots, 3-manifolds, lens spaces, homology spheres, Poincare conjecture, Geometrization conjecture, Thurston geometries	[Adams, Ch 5,9]

Homework assignments

Homework 1: due Wednesday, September 5

Homework 2: due Wednesday, September 12

Homework 3: due Monday, September 24

Homework 4: due Wednesday, October 3

Homework 5: due Friday, October 12

Homework 6: due Friday, November 9

Homework 7: due Friday, November 16

Homework 8: due Monday, December 3