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The definition of Wick rotation in curved spacetime and in quantum gravity is ambiguous. I will concentrate on a definition where the Wick rotation is seen as an operation on the metric, leaving the underlying manifold untouched. Still, if one tries to reduce the ambiguity by making some reasonable demands, it seems that in the presence of horizons one cannot maintain the global topology. I will also discuss the possibility of not performing a Wick rotation at all.