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It is shown that no purely topological condition implies the equality of thepolynomial and rational hulls of a set: For any compact subset $K$ of aEuclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that ishomeomorphic to $K$ and is rationally convex but not polynomially convex. Inaddition, it is shown that for the surfaces in ${\mathbb C}^3$ constructed byIzzo and Stout, whose polynomial hulls are nontrivial but contain no analyticdiscs, the polynomial and rational hulls coincide, thereby answering a questionof Gupta. Equality of polynomial and rational hulls is shown also for$m$-dimensional manifolds ($m\geq 2$) with polynomial hulls containing noanalytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio andWold.

No topological condition implies equality of polynomial and rational hulls