The lecture titles below approximate what might happen in each class. I will adjust them as time goes on.
Tuesday 13 January
Point-set topology
Subsets of n-space: convergence and continuity, compactness and the Heine-Borel theorem, least upper bounds on the line. Metric spaces: compactness again, open and closed sets, convergence and continuity again. Topological spaces: continuity again, products, disjoint unions, identification spaces.
Background reading: Hatcher’s notes on introductory point-set topology.
For more background reading, I really like the text General Topology by John L. Kelley, but it is a bit tricky to find a (legal) copy online. So if you’re interested, then I’ll leave you to track down a copy on your own.
Thursday 15 January
More on point-set topology
The Hausdorff separation property, maps between compact Hausdorff spaces. Countability axioms and metrizability. Connectedness, cut points, a fixed-point theorem.
Background reading: Hatcher’s notes again.
Here is a sample paper from the point-set topology era (by Moore). The article does include a lovely theorem, but overall it gives the impression of a subject collapsing under its own weight ...
Tuesday 20 January
The singular homology groups of a topological space
Simplices, standard simplices, singular simplices, free abelian groups, singular chain groups, the boundary map, singular homology. Singular homology of a point.
Background reading: Chapter One in the recommended text by Vick (see the syllabus)
First homework assigned. Please see these instructions, along with this set of sample solutions along with the TeX file used to create it (and an image file that you'll need to run TeX on the solutions file without generating an error).
Thursday 22 January
Chain complexes, chain homotopies, and homology of a convex set
First principles of homological algebra, chain homotopies, homology of a convex set.
Background reading: Chapter One in Vick (see the syllabus). These note of mine on a course in singular homology may also be useful.
Tuesday 27 January
Homotopy invariance
Homotopy for topological spaces. Homotopy invariance of singular homology. First homework due. Second homework assigned.
Background reading: Chapter One in Vick or lectures 3 and 4 of my course in singular homology.
Thursday 29 January
Homotopy invariance, concluded, and the Mayer-Vietoris sequence, begun
Background reading: Chapter One in Vick or lectures 4 and 5 of my course in singular homology.
Here is a short biography of Leopold Vietoris.
Tuesday 03 February
The Mayer-Vietoris sequence, continued
Background reading: Chapter One in Vick or lectures 4 and 5 of my course in singular homology.
Second homework due. Third homework assigned.
Thursday 05 February
First applications of singular homology
Background reading: Chapter One in Vick or lectures 7, 8 and 9 in my course in singular homology.
Tuesday 10 February
Barycentric subdivision
Third homework due. Fourth homework assigned.
Background reading: Chapter One in Vick or lecture 13 in my course in singular homology.
Thursday 12 February
Mayer-Vietoris, concluded
Background reading: Vick, Appendix I, or these new notes of mine on the barycentric subdivision operator in singular homology,
Tuesday 17 February
The Jordan-Brouwer separation theorem, part one
Fourth homework due. Fifth homework assigned.
Background reading: Chapter 1 of Vick, or Lectures 10, 11 and 12 in my course in singular homology.
Thursday 19 February
The Jordan-Brouwer separation theorem, part two
Background reading: Chapter 1 of Vick, or Lectures 10, 11 and 12 in my course in singular homology.
Tuesday 24 February
The Jordan-Brouwer separation theorem, part three
Background reading: Chapter 1 of Vick or my slightly revised lecture notes
Fifth homework due. Sixth homework assigned.
Thursday 26 February
Fundamental group
Tuesday 03 March
Class canceled today due to weather
Sixth homework due. Seventh homework assigned.
Thursday 05 March
Classification of covering spaces
Tuesday 10 March
Spring Break
Thursday 12 March
Spring Break
Tuesday 17 March
Van Kampen theorem
Seventh homework due.
Thursday 19 March
Examples and applications of covering space theory
Tuesday 24 March
Homology of a product space, part one
Eighth homework assigned.
Thursday 26 March
Homology of a product space, part two
Tuesday 31 March
Homology of a product space, part three
Eighth homework due. Ninth homework assigned.
Thursday 02 April
Cohomology, part one
Tuesday 07 April
Cohomology, part two
Ninth homework due. Tenth homework assigned.
Thursday 09 April
Duality, part one
Tuesday 14 April
Duality, part two
Tenth homework due. Eleventh homework assigned.
Thursday 16 April
Cellular homology, part one
Tuesday 21 April
Cellular homology, part two
Eleventh homework due. Twelfth homework assigned.
Thursday 23 April
Cellular homology, part three
Tuesday 28 April
Fixed point theory, part one
Twelfth homework due.
Thursday 30 April
Fixed point theory, part two