PMH1 Algebraic Topology

General Information

This page relates to the Pure Mathematics Honours course "Algebraic Topology".

Lecturer(s) for this course: Jonathan Hillman.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

The Algebraic Topology lecture notes are available online in pdf format, in two parts, Part I (Outline, Homology, the bare bones of categories - 41 pages; minor revision March 2009) and Part II (Fundamental group and Combinatorial Group Theory - 34 pages). Exercises are interspersed among the text (numbered separately in each part). (Note also that these notes lack figures; these shall be provided separately.)

The assignments shall be announced in class.

Assignment 1: do questions 5, 13, 15, 19. In Question 5 you may assume that the Euler characteristic of the torus is 0. Question 13 is hard, but needs no machinery beyond the definition of homotopy (equivalence). Question 15 is parallel to class work. Question 17 uses the result obtained in point 6 just above.

Assignment 2: do questions 23, 25, 29. Question 23 is perhaps the hardest. Show that you can use excision to compare the pair $(S^{n+k},X)$ with the disjoint union of $\mu$ copies of $(S^n\times{D^k},S^n\times{S^{k-1}})$. You may assume that if $k>1$ then $H_i(S^n\times{S^{k-1}};R)$ is isomorphic to $R$ if $i=0,n,k-1$ or $n+k-1$, and is $0$ otherwise. (A bonus mark if you can justify this assumption!) In question 29 you may need to use the fact that every finitely generated abelian group is a direct sum of cyclic groups.

Timetable