Have $\mathbb{Z}_2$ act on $\mathbb{T}^k$ by reversing the coordinates, i.e. $z_i\to z_{k-i+1}$. What is the orbit space $\mathbb{T}^k/\mathbb{Z}_2$ homeomorphic to?
Obviously the space is compact and connected but I don't know how to proceed much further. I've calculated the fundamental group as $\pi_1(\mathbb{T}^k/\mathbb{Z}_2)\cong \mathbb{Z}^{\phi_k}$ where $\phi_k=k/2$ for $k$ even and $\phi_k=(k+1)/2$ else and this was done by using some results I found on fundamental groups of orbit spaces from some quick googling.
[An AMS paper, Calculating the fundamental group of an orbit space, by Armstrong was the primary source, the main result being $\pi_1(X/G)\cong \bar{G}/N$ where the closure is taken in the topological group of homeomorphisms on $X$ and with the compact-open topology, and $N$ is the smallest normal subgroup containing the path component of the identity and each element of $G$ which has fixed points. The result holds for sufficiently nice $X$, (amongst other things, being simply connected is the crucial assumption), a weak path lifting assumption on $X\to X/G$ and two other simpler properties. He also briefly outlines (and provides references) how to apply the result when $X$ is not simply connected but has a universal covering space, which is the track I followed for $\mathbb{T}^k$.]
I believe I'm correct and it at least coincides with what I know for $k=2$: the orbit space is homeomorphic to the Mobius strip which has fundamental group $\mathbb{Z}$ and this is equal to $\mathbb{Z}^{\phi_2}$ as per the definition of $\phi_k$. However I dont know anything besides what I've mentioned about the orbit space for $k\geq 3$.
a) With respect to just trying to calculate the fundamental group of the orbit space: was this overkill? Are there simpler arguments? uhh.. more importantly, is this correct? I'm obviously not going to ask one to verify my work (since I barely outlined it), but if this space is at all well studied someone might recall its fundamental group...
b) What other data (topological or algebraic) do I want to look at that might help me get closer to finding out the given orbit space up to a homeomorphism and what are some ways to proceed? I started learning orientability as a place to start... I admittedly got caught up in computing the fundamental group I forgot how such is more useful for determining two spaces to not be homeomorphic to each other. If you know what the space is, a suggestion for me to figure it out myself would be appreciated, or if you can't resist, a sketch of your proof.
Thanks.