University of Regina MATH 442/842 - Algebraic Topology MATH 442/842 - Algebraic Topology Homework 5 Martin Frankland Winter 2021 Due date: Wednesday March 10, by 11:59 PM, on UR Courses. Guidelines. Group work is allowed. However, you may not look at other students' solutions. You must write your own solutions. Strikingly similar solutions will be flagged as suspicious. Please write the names of people with whom you have discussed the homework. Some similarity will then be accepted. Cite any sources that you consult. Please show your work. No credit for answers without justification. Definition. A (continuous) map f : X ! Y is a local homeomorphism if for every point x 2 X, there is an open neighborhood U ⊆ X of x such that f(U) ⊆ Y is open in Y and the =∼ restriction fjU : U −! f(U) is a homeomorphism. For example, every covering space is a local homeomorphism, but not conversely. Definition. A space X is locally path-connected if for every point x 2 X and neighbor- hood V of x, there is an open neighborhood U of x satisfying U ⊆ V and such that U is path-connected. In other words, x has arbitrarily small path-connected neighborhoods. ©2021 Martin Frankland All Rights Reserved 1 University of Regina MATH 442/842 - Algebraic Topology Problem 1. Let p: Xe ! X be a local homeomorphism. (a) If X is locally path-connected, show that Xe is locally path-connected. (b) Assume moreover that p is surjective. If Xe is locally path-connected, show that X is locally path-connected. Problem 2. Let X and Y be spaces, and let p: X × Y ! X be the projection onto the first factor. (a) Show that p has the path lifting property. (b) Show that p has the unique path lifting property if and only if the path components of Y consist of single points. (This holds in particular when Y is discrete.) Problem 3. Let X be a path-connected and locally path-connected space. If every element 1 of the group π1(X) has finite order, show that every map f : X ! S is null-homotopic. ©2021 Martin Frankland All Rights Reserved 2.