  Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by
degeneracy loci measuring the spitting type of the vector bundle restricted to
each fiber. The classes of these degeneracy loci in the Chow ring or cohomology
ring of the base are natural invariants characterizing the degenerations of the
vector bundle. When these degeneracy loci occur in the expected codimension, we
find their classes. This yields universal formulas for degeneracy classes in
terms of naturally arising vector bundles on the base. Our results hold over
arbitrary fields of any characteristic.
