We first classify singular fibers of proper stable maps of –dimensional manifolds with boundary into surfaces. Then we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with boundary.
"Singular fibers of stable maps of $3$–manifolds with boundary into surfaces and their applications." Algebr. Geom. Topol. 16 (3) 1379 - 1402, 2016. https://doi.org/10.2140/agt.2016.16.1379