James R. Munkres "Topology", 2nd edition, Prentice Hall, 2000.

Week

Material covered (and corresponding sections of Munkres)

Homework

Jan 11-15

Topological spaces, closed sets, bases (12, 13, 17)
Interior, closure, neighbourhoods, limit points, boundary, Hausdorffness (17)
Metric spaces, finer/coarser, continuous/open/closed maps (20, 21, 12, 18)

Homework assignment 1.
Due Wednesday 20th January.
Solutions 1.
Correction to problem 1.

Jan 18-22

Subspaces, homeomorphism, manifolds (16, 18, 36)
Product topology, quotient topology (15, 19, 22)

Homework assignment 2.
Due Wednesday 27th January.
Solutions 2.

Jan 25-29

Adjunction spaces, attaching cells (pg 224, sections 72, 74)
Animation of the Roman surface
Cut-and-paste techniques (74, 76), quotients by group actions
Connectedness (23, 24)

Homework assignment 3.
Due Wednesday 3rd February.
Solutions 3.

Feb 1-5

Path-connectedness and local connectedness (24, 25)
Compactness (26, 27, 28)

Homework assignment 4.
Due Friday 12th February.
Solutions 4.

Feb 8-12

Local compactness (29)
Triangulations of manifolds (78)
Classification of closed surfaces (76, 77)

Midterm 1.
Due Wednesday 17th February.
Midterm 1 solutions.

Feb 15-19

Homotopy and path homotopy (51)
The fundamental group (52)
Retractions (55)
Homotopy equivalence (58)

Homework assignment 5.
Due Wednesday 24th February.
Solutions 5.

Feb 22-26

Covering spaces (53)
Path lifting and the fundamental group of the circle (54)
Brouwer fixed-point theorem (55)
The fundamental group of S^n (59)

Homework assignment 6.
Due Wednesday 3rd March.
Solutions 6.

Mar 1-5

Direct sums, free products, free groups (67, 68, 69)
Seifert-van Kampen theorem (70)
Wedges of circles and adjoining two-cells (71, 72)
Fundamental groups of surfaces (74, 75)

Homework assignment 7.
Due Wednesday 17th March.
Solutions 7.

Mar 8-12

Spring break - no classes

Mar 15-19

Covering spaces and general lifting (54, 79)
The universal covering space (80)
Tessellations of the hyperbolic disc by squares and hexagons
(For the genus two surface we get a tessellation by octagons)

Homework assignment 8.
Due Wednesday 24th March.
Solutions 8.

Mar 22-26

Existence of covering spaces (82)
Covering transformations (81)
Tessellation of S^3 by dodecahedra: one, two, three

Midterm 2.
Due Wednesday 31st March.
Midterm 2 solutions.

Mar 29-Apr 2

DeRham cohomology on R^n

Homework assignment 9.
Due Wednesday 7th April.
Solutions 9.

Apr 5-9
Extra class Fri 10am

DeRham cohomology on manifolds
Poincare Lemma
Mayer-Vietoris sequence

Homework assignment 10.
Due Wednesday 21st April.
Solutions 10.

Apr 12-16

No classes this week

Apr 19-23

Integrating one-forms in R^2
The homotopy axiom
H^1(M) and the fundamental group

Homework assignment 11.
Due Wednesday 28th April.
Solutions 11.

Apr 26-30

DeRham cohomology of surfaces
Final exam 8am Friday 30th