Rotations of the two-sphere and their Lefschetz number

The Lefschetz number is $2$. Every rotation is homotopic to the identity through rotations. Since $H_0(S^2;\mathbb{Q})=H_2(S^2;\mathbb{Q})=\mathbb{Q}$ and the other groups vanish by [F1], the trace formula gives $L(\mathrm{i}\mathrm{d})=1+1=\chi (S^2)=2$, and homotopy invariance gives $L(R)=2$. Equivalently, $R_{\ast }$ is the identity on $H_0$ and multiplication by $\mathrm{deg}R=1$ on $H_2$, so $L(R)=1+1$.

The fixed points of a generic rotation. In coordinates $z$ on $S^2$ centred at the north pole the rotation is $z\mapsto e^{i\theta }z$ for $\theta \mathrm{\notin }2\pi \mathbb{Z}$; fixed points solve $(e^{i\theta }-1)z=0$, so $z=0$, and near the south pole the same computation in the chart $w=1\mathrm{/}z$ shows that only the south pole is fixed. At each pole the displacement has invertible differential $1-e^{\pm i\theta }\mathrm{\ne }0$, so both fixed points are nondegenerate and by [F2] each has index $\mathrm{sign}\det (1-e^{\pm i\theta }\cdot {\mathrm{i}\mathrm{d}}_{{\mathbb{R}}^2})=+1$ (the linear map is a positive multiple of a rotation). Hence $I(R)=1+1=2=L(R)$, in agreement with the index formula. For the half-turn $\theta =\pi$, the same two poles are the entire fixed set and $I-DR=2I$ on each tangent plane, so each is nondegenerate of index $+1$. For $\theta \in 2\pi \mathbb{Z}$ the map is the identity and fixes all of $S^2$; this fixed set is not isolated, so only the Lefschetz number $2$ is asserted directly.