#### Volume 16, issue 3 (2016)

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Singular fibers of stable maps of $3$–manifolds with boundary into surfaces and their applications

### Osamu Saeki and Takahiro Yamamoto

Algebraic & Geometric Topology 16 (2016) 1379–1402
##### Abstract

We first classify singular fibers of proper ${C}^{\infty }$ stable maps of $3$–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.

##### Keywords
stable map, singular fiber, manifold with boundary, cobordism
##### Mathematical Subject Classification 2010
Primary: 57R45
Secondary: 57R35, 57R90, 58K15, 58K65