I am interested when manifolds $M$ (with boundary) are displaceable, i.e be pushed off themselves. Concretely, I mean there is a diffeomorphism $\phi: M \rightarrow M$ that is identity on the boundary such that $\phi(M_0) \cap M_0 = \emptyset$; here $M_0 \subset M$ is the result of removing a small collar neighborhood of $\partial M$ from $M$. So $M_0$ and $M$ are diffeomorphic. For manifolds to be displaceable, the intersection form needs to vanish.
For example, suppose that $M^{2n}$ is a 2n-manifold with a handlebody decomposition with handles of index at most n. Also, suppose that the intersection form of $M^{2n}$ vanishes. Is it true that $M^{2n}$ is displaceable?
The following manifolds are displaceable: $M = \mathbb{R}^{2n}, \mathbb{R}^2 \times W^{2n-2}, T^*S^k \times W^{2n-2k}$ (for k odd).