I haven’t updated this blog in a while, due to qualifying exam preparation and research projects. I’ll write about the latter soon. While studying for the geometry/topology qual, I asked a basic question: Is path connectedness a homotopy invariant? Turns out the answer is yes, and I’ve written up a quick proof of the fact below. You can view a pdf of this entry here.
Proposition 1 Let be a homotopy equivalence. If is path connected, then so is .
Proof: It’s clear that the image of is path connected. Thus, it is enough to show that any point of can be connected to a point of . Let be a map such that is homotopic to , via the homotopy . Let . Then, and is a path from to .
We can deduce the following result from the proof of proposition 1.
Corollary 2 Let be a homotopy equivalence. Then is path connected if, and only if, is path connected.
Proof: If is path connected, then is path connected. Hence is path connected.