Give examples of two $C_2$ actions on $S^n$ such that the orbit spaces are not homotopy equivalent.
My attempt:
For $n \ge 2$. I considered the following actions of $C_2$ on $S^n$ : (i) mapping to antipodal points , (ii) the trivial action.
Action (i) yields the Orbit space to be $\Bbb RP^n$ and for action (ii) we have the Orbit space to be $S^n$ itself!
$$\forall n \ge 2,\pi_1(\Bbb RP^n)= \Bbb Z/2 \Bbb Z, \pi_1(S^n)= 0$$
So done, $\forall n \ge 2$.
But this argument doesn't work for $n=1$, since, $\Bbb RP^1 \cong S^1$
So how to deal with $n=1?$
Thanks in advance for help!
Consider $C_2$ acts by reflection across an equatorial $S^{n-1}$ of $S^n$. This gives the orbit space $D^n\simeq pt$.
Of course, for $n=0$ $S^0$ is just two points and the two different actions of $C_2$ have different cardinality of quotient.