# Correction to: Geometry and topology of the Kerr photon region in the phase space

• Carla Cederbaum
• Sophia Jahns
Correction

## 1 Correction to: General Relativity and Gravitation (2019) 51:79  https://doi.org/10.1007/s10714-019-2561-y

The proof of Theorem 10 can be considerably simplified, as was pointed out to us by Gregory J. Galloway: Indeed, it is unnecessary to rule out L(2; 0); since 0 and 2 are not coprime, this case does not need to be considered, which makes it superfluous to argue that $$P_0$$ admits a Seifert fibration without exceptional fibers (Proposition 15). Thus, one is left with the case $$P_0\approx L(2;1)\approx SO(3)$$.

From a slightly different point of view, one may also argue as follows: Since $$P_0$$ is a closed 3-dimensional manifold with $$\pi _1(P_0)=\mathbb Z_2$$, it is doubly covered by an $$\mathbb S^3$$ (by the Poincaré conjecture). By the elliptization conjecture, this $$\mathbb S^3$$ can be taken to be the standard 3-sphere and the group $$\mathbb Z_2$$ as a subgroup of SO(3) acting on it. (For the statements of the Poincaré and the elliptization conjecture, see  [4, 5]; for the proofs covering these conjectures see [1, 2, 3].) Hence, $$P_0$$ is the quotient $$\mathbb S^3 /\mathbb Z_2 \approx \mathbb R P^3\approx SO(3)$$.

Recalling how $$P_0$$ was obtained as a slice $$P\cap \{t=0, p_0=-1\}$$ of the photon region in the phase space, this proves Theorem 10.

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