## Obstructions for $$\pi_1$$ of the space of pseudo-isotopies of disks in non-stable range
We study the fundamental group of the space of pseudo-isotopies $$\mathcal{C}(D^d)$$ of odd-dimensional disks by using parametrized Morse theory. In particular we introduce some algebraic obstructions for trivializing pseudo-isotopy fiber bundles over $$S^2$$ and show that if the obstructions for an element $$\xi$$ of $$\pi_1\mathcal{C}(D^d)\otimes\mathbb{Q}$$ vanish, then $$\xi$$ is trivial when $$d\geq 7$$ odd, and $$\xi$$ can be represented by a family of framed embeddings of 2-spheres in $$\mathbb{R}^5$$ and some simple pattern parametrized by some abelian group when $$d=5$$.