# Singular Fibers of Stable Maps of Manifold Pairs and Their Applications

## Abstract

Let (*M*, *N*) be a manifold pair, where *M* is a closed 3-dimensional manifold and *N* is a closed 2-dimensional submanifold of *M*. In this paper, we first classify singular fibers of \(C^\infty \) stable maps of (*M*, *N*) 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 manifold pairs \((M', N')\), where \(M'\) is a closed surface and \(N'\) is a closed 1-dimensional submanifold of \(M'\). We also give the 2-colored versions of all these results, when the submanifold separates the ambient manifold into two parts.

## Keywords

Manifold pair Stable map Singular fiber Cobordism 2-coloring## 2000 Mathematics Subject Classification

Primary 57R45 Secondary 57R35 57R90 58K15 58K65## Notes

### Acknowledgements

The authors would like to thank the anonymous referee for some comments which improved the presentation of the paper. The first author has been supported in part by JSPS KAKENHI Grant Numbers JP23244008, JP23654028, JP25540041, JP15K13438, JP16H03936, JP16K13754, JP17H01090, JP17H06128. The second author has been supported in part by JSPS KAKENHI Grant Numbers JP23654028, JP15H03615, JP15K04880, JP15K13438, JP17H01090, JP17H06128.

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