Does every involution have a fixed point?

Question

I was trying to find functions $f:X\to X$ such that $f^2$ is the identity on $X$ and such that $f(x)\ne x$ for all $x\in X$. First, this is a bit vague, of course you can take any two-element set $X=\{a,b\}$ with $f(a)=b$ and $f(b)=a$ but bear with me.

An "operation" $f$ on $X$ such that $f^2$ is the identity is, to my eye, a "symmetrical" operation, like rotation or reflection of the plane/space or $n\mapsto -n$ in $\mathbb{Z}$ (or $a\mapsto a^{-1}$ in any group $G$). All of this that I could think of have an axis of reflection/symmetry or center of rotation that remains fixed. The valid examples that came to minde were like $n\mapsto -n$ on $\mathbb{Z}\setminus\{0\}$ and similar. All of them somewhat "unnatural".

So my question is 1) Can this question be made more precise/rigorous? and 2) Are there naturally occurring examples of involutions $f:X\to X$ where $X$ has some additional (algebraic) structure?

I say algebraic because half a turn on a (geometrical/topological) annulus does the trick and you may say that is natural. Maybe we should require $X$ to be simply connected.

For a concrete question: find $f:X\to X$ with $X$ simply connected such that $f^2=id_X$, $f$ has no fixed point and $f$ comes up naturally

Thanks!

Answer 1

The Lefschetz fixed point theorem can be used to prove the following.

Claim: Let $X$ be a compact triangulable space (e.g. most compact manifolds, with or without boundary) whose Euler characteristic $\chi(X)$ is odd. Then every involution $f : X \to X$ has a fixed point.

This is a nice exercise and we don't actually need the proof here so I'll leave it as an exercise.

This means that to find an involution with no fixed point we should restrict our attention to spaces whose Euler characteristic is even. An easy connected example is the circle $S^1$: a $180^{\circ}$ rotation has no fixed points. Of course $S^1$ is not simply connected, but every sphere has even Euler characteristic and is simply connected except for $S^1$ and $S^0$, and in fact every sphere is an example, with $f$ the antipode map. The $3$-sphere $S^3$ even has the additional structure of being a Lie group (with $3$ different names!), namely the special unitary group $SU(2)$.

If $X$ is connected and $f : X \to X$ is an involution with no fixed point then (under some mild additional hypotheses) the quotient by $f$ defines a space $Y$ of which $X$ is a nontrivial $2$-fold covering space; for the spheres above these quotient spaces are the real projective spaces $\mathbb{RP}^n$, so this example is quite important / natural.

Conversely, every connected space $Y$ with a nontrivial $2$-fold covering space produces an example of a connected space with an involution with no fixed points. Taking this $2$-fold cover doubles the Euler characteristic, which gives some intuition for the claim above. (Everywhere I need some mild hypotheses so that covering space theory works out and Euler characteristics exist.) A connected space $Y$ has a nontrivial $2$-fold covering space iff its fundamental group $\pi_1(Y)$ has a subgroup of index $2$ iff it admits a surjective homomorphism to the cyclic group $C_2$. The real projective spaces above have fundamental group exactly $C_2$ so the spheres arise in this way, again quite naturally.