## MA 651: Topology II

Kate Ponto
Spring 2020

Syllabus

### Announcements

It looks like subgroups of free groups are free might not be in Lee's book. Alternative references are:
Hatcher section 1.A
Munkres chapter 14

Office hours will be M 11-12, T 3:30-4:30 and F 1:30-2:30.

The primary textbook for this class will be Introduction to Topological Manifolds by Lee. (It is available through SpringerLink - check the library website!)

Algebraic Topology by Hatcher and Topology by Munkres can also be useful references.

### Homework

(Pages from Lee)
1/22 Ex 7.6, a careful homotopy for Th. 7.11c, P. 7-1, 7-3, 7-10 183-195, 197-199
1/29 7-4, 7-5, 7-2, 7-6 199-205, 205-208, 217-220
2/5 7-9, 7-11, 7-19, 8-1 (you can use the fact that the fundamental group of the circle is non trivial.), 8-7 220-224, 225-229
2/12 8-4, 8-11, 8-6, 8-8 277-283
2/19 11-2, 11-4, 11-12, 11-13, 11-14 283-287, 287-291
2/26 12-2c, 12-3 292-297
3/4 11-16, 11-20, 11-17 (you can replace the space by the wedge of S1 and S2 if you want), 11-18 297-302
3/11

11-19, 12-4, 12-15 (you can use Theorem 12.18, 6.16, 6.17), 12-12 307-311, 311-322, 233-244
3/25 9-1, 9-4, 10-1, 10-2, 10-5, 10-9 251-257, 261-264, 264-273
4/1 9-5, 10-6, 10-21, 10-7, 10-13
4/8 10-3, 10-20, 10-19, Compute some homology 339-347
4/15 Compute the simplicial homology of the Klein bottle and sphere, Practice with exactness 347-355
4/22 13-2, 13-6, 13-3, 13-7, Show that the quotient map S^1\times S^1\to S^2 collapsing the subspace S^1 \vee S^1 to a point is not nullhomotopic by showing that it induces an isomorphism on H_2. On the other hand, show via covering spaces that any map S^2 \to S^1\times S^1 is nullhomotopic. (You can use the fact that X\cup C(A) is homotopy equivalent to X/A if X and A are CW complexes.) 355-363
4/29 Homology of spaces

Now three problems!

363-366, 369-374