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Let $\Sigma$ be an oriented compact hypersurface in the round sphere$\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case ofthe torus, $\Sigma$ is further assumed to be contained in a contractible subsetof $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer$N$ there exists an eigenfunctions $\psi$ of the Laplacian on $\mathbb{S}^n$ or$\mathbb{T}^n$ satisfying $\Delta \psi=-\lambda \psi$ (with $\lambda=N(N+n-1)$or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with aconnected component of the nodal set of $\psi$ given by~$\Sigma$, up to anambient diffeomorphism.

High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology