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    Action of $\pi_1$ on $\pi_n$ of $\mathbb{RP}^n$

    E-HAT-4.2-2

    Area
    Topology
    Topics
    Higher Homotopy Groups
    Status
    draft

    Show the action of \(\pi_1(\mathbb{RP}^n)\) on \(\pi_n(\mathbb{RP}^n) \approx \mathbb{Z}\) is trivial for \(n\) odd and nontrivial for \(n\) even.

    Solution

    For \(n\ge2\), the universal covering is \[ p:S^n\to\mathbb{RP}^n, \] with deck group \(\mathbb Z_2\) generated by the antipodal map \[ a:S^n\to S^n, \qquad a(x)=-x. \] Under the canonical identification \[ \pi_n(\mathbb{RP}^n)\cong\pi_n(S^n)\cong\mathbb Z, \] the \(\pi_1\)-action is the action induced by this deck transformation. The antipodal map has degree \[ \deg a=(-1)^{n+1}. \] Hence its action on \(\pi_n(S^n)\cong\mathbb Z\) is multiplication by \((-1)^{n+1}\).

    Thus when \(n\) is odd the action is multiplication by \(+1\), hence trivial, while when \(n\) is even it is multiplication by \(-1\), hence nontrivial: \[ \boxed{ \gamma\cdot m= \begin{cases} m,&n\text{ odd},\\ -m,&n\text{ even}. \end{cases}} \]

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