## Introductory Topology, Spring 2020 ### Homework:

Assignment 1 (due 1/17): Section 13, problems 1, 3, 7, 8; Section 16, problems 1, 2, 6; Section 17, problems 5, 6, 7, 19 (these will accessible once we learn about the closure and interior of a subset on Monday, or just quickly read the definition!)

Assignment 2 (due 1/24): Section 16, problems 4, 8; Section 17, problem 13; Section 18, problems 2, 5, 10, 13; Section 19, problems 3, 7, 8; Section 22, Supplementary Exercises on Topological Groups, problems 1, 3; also, if you’re unfamiliar with the topology of metric spaces, read Sections 20 and 21

Assignment 3 (due 1/31): Section 23, problems 2, 9, 10, 11; Section 24, problems 1, 2, 3, 8, 9, 10

Assignment 4 (due 2/7): Section 25, read it, then solve problems 1, 4, 5, 9; Section 26, problems 4, 5, 7, 8, 13; Section 27, problems 3, 4; Section 30, problems 5, 16

Assignment 5 (due 2/14): Section 26, problem 13; Section 29, problems 1, 6, 10 (you’ll need to read about locally compact spaces and 1-point compactification, which are in that section); Section 30, problem 8; Section 31, problems 5, 8; Section 32, problems 4, 5; Section 33, problem 2; Section 35, problems 4, 7

Assignment 6 (due 2/21): Section 44, problems 1,2; Section 51, problems 2, 3; Section 52, problems 1, 3, 4, 5, 7

Assignment 7 (due 3/20): Section 53, problems 3, 6; Section 54, problems 1, 5, 8; Section 57, problem 2 and read the proof of Thm 57.4; Section 58, problems 2, 8, 9; Section 59, problems 1, 3

Assignment 8 (due 4/4): Section 72, read it + problems 1, 2; Section 79, problems 1, 2 (+ read the relevant parts of pages 372-373), 7; Page 79 in Hatcher, problems 1, 2, 3, 7

Assignment 9 (due 4/11): Hatcher Section 1.3, problems 4, 6, 10, 14, 16, 23; Munkres Section 79, problem 4 and Section 81, problem 5

Assignment 10 (due 5/1): Section 75, problem 2 (for now, H_1(X) for X path-connected just means the abelianization of the fundamental group); Section 78, problem 1 (a labeling scheme just tells you which edges to identify, say as you travel around each triangle counterclockwise); Section 81, problems 1 and 2; Lastly, prove that if E and B are connected closed orientable surfaces and E is a covering space of B, then we must have genus(E) = 1+n(genus(B)-1), where n is the number of sheets of the cover. As an example, try to construct a three sheeted cover of a genus 2 surface by a genus 4 surface (here’s a hint for the last bit: try to construct a Z/3-action on a genus 4 surface. Do ask if you want a hint!)