Algebraic topology homework problems

You can read answers to these homework problems, written by Christopher Walker. The course used this book: James Munkres, Topology, 2nd edition, Prentice Hall, 1999. algebraic topology homework problems winter quarter 2011 3 (12) The Klein bottle KB is the quotient space obtained from the square I 2 via the boundary identications (0, y) (1, 1 y) and (x, 0) (x, 1). Problem sets are due on Tuesdays at the beginning of class, except as noted below. If you can't make it to class, put it in my mailbox before class.

If you can't make it to class, put it in my mailbox before class. Problem 5. Let A; B; C, and D be sets. Suppose f: A! C and g: B! D are homeomorphisms. Use f and g to de ne a homeomorphism between A B and C D, and prove that it is a bijection (i. e. you do NOT need to show continuity. ) Problem 6. Give a geometric description of a homeomorphism from the sphere S2 to the unit cube C f(x; y; z) 2 Solutions to Homework# Algebraic topology homework problems Hatcher, Chap.

0, Problem 4. Denote by iA the inclusion map A, ! X. Consider a homotopy F: X I! Solutions to Homework# 2 From the properties of quotient topology we deduce that j is a homeomorphism. Here is the promised extra credit assignment. The problems are from chapter 0 of Hatcher's book: Chapter 0# 11, 15, 16, 18, 19, 24; Hatcher's Book Here is a link to Hatcher's book on algebraic topology: Hatcher, Algebraic Topology; This link points to the doublepage version.

Problem 1. Find an example of a topological space X that is not Haussdorf such that each point in X has a neighborhood homeomorphic to the open interval (1, 1). Problem 2. Show that the gure 8 (viewed as a subset of the plane, with a topology induced from the usual topology of IR2) is not homeomorphic to a circle. Problem 3. Algebraic Topology Homework 4 Solutions Here are a few solutions to some of the trickier problems Recall: Let Xbe a topological space, A Xa subspace of Algebraic Topology, Math 4152b9052b (and York!

), Winter 2016 Algebraic topology is the study of topological spaces using tools of an algebraic nature, such as homology groups, cohomology groups and homotopy groups. I plan a long list of suggested problems for your Homework& Problem Session; these will be in addition to the textbook problems.

We will have a weekly hour long problem session. We will have a weekly hour long problem session.