Questions tagged [involutions]

I was wondering if the following theorem still holds if one only requires $\tau$ to be a homeomorphism and not a diffeomorphism: "Let M be a smooth manifold and $\tau:M\to M$ a differentiable function such that $\tau(\tau(x))=x$ and $\tau(x)\neq x$.…

Let $_$ be the number of involutions in $\sigma_n$. Show that $_0 = _1 = 1$ and for $ \geq 2$ $_ = _{−1} + ( − 1)_{−2}.$

Understand that involution means to multiply value with the same value so many times as involution number. So, understand that $1*1=1$ $t^2 * t^2$ would be 2t^2 (sorry, now understand that this is incorrect. Should be $t^{(2+2)}$ because rule is…