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We consider the singularly perturbed semilinear parabolic problem  $u_t-d^2\Delta u+u=f(u)$ with homogeneous Neumann boundary  conditions on a smoothly bounded domain $\Omega\subseteq \mathbb{R}^N$.  Here $f$ is superlinear at $0$, and $\pm\infty$ and has subcritical  growth. For small $d>0$ we construct a compact connected invariant set $X_d$  in the boundary of the domain of attraction of the asymptotically  stable equilibrium $0$. The main features of $X_d$ are that it  consists of positive functions that are pairwise non-comparable,  and its topology is at least as rich as the topology of  $\partial\Omega$ in a certain sense. If the number  of equilibria in $X_d$ is finite, then this implies the existence of  connecting orbits within $X_d$ that are not a consequence of a well  known result by Matano.

An invariant set generated by the domain topology for parabolic semiflows with small diffusion