Involution and Covering space

Question

Is there a connected topological space such that admits a free involution, trivial fundamental group and furthermore has the set of real number as it's covering space?

Answer 1

A simply connected topological space is its own universal cover, so your question reduces to the following:

Does $\mathbb{R}$ admit a free involution $f$?

I'm going to assume you want the involution $f$ to be continuous. If you don't require continuity, then there are many examples

If $f$ has no fixed points, it is injective and is therefore either increasing or decreasing. If it were increasing, we'd have $x < f(x) < f(f(x)) = x$ which is a contradiction. If it were instead decreasing, we would find that $x = f(f(x)) < f(x) < x$ which is again a contradiction.

Therefore, every continuous involution of $\mathbb{R}$ has a fixed point. In fact, it either has exactly one fixed point, or it is the identity map; see this answer.