singular homological degree of a reflection

Given the n-sphere and the \(k\)-th reflection, the singular homological degree is

\begin{align*} \mathrm{deg}(r_k) = (-1)^k \end{align*}

w.l.o.g. for \(r_{2}\) as the general case reduces to

\begin{align*} \tau_{k,2} r_{k} \tau_{k,2}^{-1} = r_2 \end{align*}

where

\begin{align*} \tau_{k,2}: \mathbb{R}^{n+1} \rightarrow& \mathbb{R}^{n+1} \\ (x_1,x_2,...,x_k,...,x_{n+1}) \mapsto& (x_1,x_k,...,x_2,...,x_{n+1}) \end{align*}

(cf. singular homological degree invariant under conjugation)

Here

\begin{align*} r_2 = \Sigma^{n-1}(\overline{}) \end{align*}

is the \(n-1\)-fold suspension where

\begin{align*} \overline{}: S^2 \rightarrow S^2 \\ (x_1,x_2) \mapsto& (x_1, - x_2) \end{align*}

is the complex conjugation

Then <>