# 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.

## References

- 1.Damon, J.: The unfolding and determinacy theorems for subgroups of \(\cal{A}\) and \(\cal{K}\). Mem. Amer. Math. Soc.
**50**(306) (1984), x+88 ppGoogle Scholar - 2.Martins, L.F., Nabarro, A.C.: Projections of hypersurfaces in \({\mathbb{R}}^4\) with boundary to planes. Glasg. Math. J.
**56**, 149–167 (2014). https://doi.org/10.1017/S001708951300013X - 3.Ohmoto, T.: Vassiliev complex for contact classes of real smooth map-germs. Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem.
**27**, 1–12 (1994)Google Scholar - 4.Saeki, O.: Topology of singular fibers of differentiable maps. Lect. Notes in Math. vol. 1854. Springer, Berlin (2004)Google Scholar
- 5.Saeki, O.: Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs. Algebr. Geom. Topol.
**6**, 539–572 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Saeki, O., Yamamoto, T.: Singular fibers of stable maps and signatures of \(4\)-manifolds. Geom. Topol.
**10**, 359–399 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Saeki, O., Yamamoto, T.: Singular fibers and characteristic classes. Topol. Appl.
**155**, 112–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Saeki, O., Yamamoto, T.: Co-orientable singular fibers of stable maps of \(3\)-manifolds with boundary into surfaces. Sūrikaisekikenkyūsho Kōkyūroku, No. 1948, 137–152 (2015)Google Scholar
- 9.Saeki, O., Yamamoto, T.: Singular fibers of stable maps of \(3\)-manifolds with boundary into surfaces and their applications. Algebr. Geom. Topol.
**16**, 1379–1402 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Saeki, O., Yamamoto, T.: Cobordism group of Morse functions on surfaces with boundary. In: Real and Complex Singularities. Contemporary Mathematics, Proceedings of the XIII International Workshop on Real and Complex Singularities, São Carlos, 2014, vol. 675, pp. 279–297 (2016)Google Scholar
- 11.Shibata, N.: On non-singular stable maps of \(3\)-manifolds with boundary into the plane. Hiroshima Math. J.
**39**, 415–435 (2000)MathSciNetzbMATHGoogle Scholar - 12.Vassilyev, V.A.: Lagrange and Legendre characteristic classes, Translated from the Russian, Advanced Studies in Contemporary Mathematics, vol. 3. Gordon and Breach Science Publishers, New York (1988)Google Scholar
- 13.Wall, C.T.C.: Cobordism of pairs. Comment. Math. Helv.
**35**, 136–145 (1961)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Yamamoto, T.: Classification of singular fibres of stable maps of \(4\)-manifolds into \(3\)-manifolds and its applications. J. Math. Soc. Jpn.
**58**, 721–742 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Yamamoto, T.: Euler number formulas in terms of singular fibers of stable maps, in “Real and complex singularities”. World Sci. Publ., Hackensack, NJ, pp. 427–457 (2007)Google Scholar
- 16.Yamamoto, T.: Singular fibers of two-colored maps and cobordism invariants. Pac. J. Math.
**234**, 379–398 (2008)MathSciNetCrossRefzbMATHGoogle Scholar