Path connectedness is a homotopy invariant

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 {f : X \rightarrow Y} be a homotopy equivalence. If {X} is path connected, then so is {Y}.

Proof: It’s clear that the image of {f} is path connected. Thus, it is enough to show that any point of {Y} can be connected to a point of {f(X)}. Let {g : Y \rightarrow X} be a map such that {f\circ g} is homotopic to {\text{id}_Y}, via the homotopy {h : Y \times I \rightarrow Y}. Let {y \in Y}. Then, {y' = f(g(y)) \in f(X)} and {\gamma(t) = h(y, t)} is a path from {y'} to {y}. \Box

We can deduce the following result from the proof of proposition 1.

Corollary 2 Let {f : X \rightarrow Y} be a homotopy equivalence. Then {X} is path connected if, and only if, {f(X)} is path connected.

Proof: If {f(X)} is path connected, then {Y} is path connected. Hence {X} is path connected. \Box