The homology fiber bundle smoothness conjecture for projective morphisms

Let $Y$ be a complex analytic manifold and let $f:Y\to\Delta$ be a projective morphism, where $\Delta$ is a disc. A morphism is a homotopy fiber bundle if it has the homotopy-local triviality property, and a $\mathbb{Z}$-homology fiber bundle if its fibers have the corresponding integral homology-local triviality property. The homology fiber bundle smoothness conjecture. The following conditions are equivalent:

1. $f$ is smooth, hence a differentiable fiber bundle.
2. $f$ is a homotopy fiber bundle.
3. $f$ is a $\mathbb{Z}$-homology fiber bundle.

The conjecture was proposed because all previously known examples of homotopy or homology fiber bundles that were not topological fiber bundles had singular total spaces. The paper's abstract states that it constructs counterexamples with smooth total space, thereby disproving this conjecture.