## Algebraic & Geometric Topology

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

#### Abstract

We first classify singular fibers of proper $C∞$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1379-1402.

Dates
Revised: 18 June 2015
Accepted: 21 September 2015
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895850

Digital Object Identifier
doi:10.2140/agt.2016.16.1379

Mathematical Reviews number (MathSciNet)
MR3523043

Zentralblatt MATH identifier
1360.57034

#### Citation

Saeki, Osamu; Yamamoto, Takahiro. Singular fibers of stable maps of $3$–manifolds with boundary into surfaces and their applications. Algebr. Geom. Topol. 16 (2016), no. 3, 1379--1402. doi:10.2140/agt.2016.16.1379. https://projecteuclid.org/euclid.agt/1511895850

#### References

• J Damon, The relation between $C\sp{\infty }$ and topological stability, Bol. Soc. Brasil. Mat. 8 (1977) 1–38
• C,G Gibson, K Wirthmüller, A,A du Plessis, E,J,N Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics 552, Springer, Berlin (1976)
• M Golubitsky, V Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics 14, Springer, New York (1973)
• L,F Martins, A,C Nabarro, Projections of hypersurfaces in $\mathbb{R}\sp 4$ with boundary to planes, Glasg. Math. J. 56 (2014) 149–167
• T Ohmoto, Vassiliev complex for contact classes of real smooth map-germs, Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. 27 (1994) 1–12
• A du Plessis, T Wall, The geometry of topological stability, London Mathematical Society Monographs, New Series 9, Clarendon Press, New York (1995)
• O Saeki, Topology of singular fibers of differentiable maps, Lecture Notes in Mathematics 1854, Springer, Berlin (2004)
• O Saeki, Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs, Algebr. Geom. Topol. 6 (2006) 539–572
• O Saeki, S Takahashi, Visual data mining based on differential topology: a survey, Pac. J. Math. Ind. 6 (2014) Article 4
• O Saeki, S Takahashi, D Sakurai, H-Y Wu, K Kikuchi, H Carr, D Duke, T Yamamoto, Visualizing multivariate data using singularity theory, from: “The impact of applications on mathematics”, (M Wakayama, R,S Anderssen, J Cheng, Y Fukumoto, R McKibbin, K Polthier, T Takagi, K-C Toh, editors), Math. Ind. (Tokyo) 1, Springer, Tokyo (2014) 51–65
• O Saeki, T Yamamoto, Singular fibers of stable maps and signatures of $4$–manifolds, Geom. Topol. 10 (2006) 359–399
• O Saeki, T Yamamoto, Singular fibers and characteristic classes, Topology Appl. 155 (2007) 112–120
• O Saeki, T Yamamoto, Co-orientable singular fibers of stable maps of $3$–manifolds with boundary into surfaces, Sūrikaisekikenkyūsho Kōkyūroku 1948 (2015) 137–152
• N Shibata, On non-singular stable maps of $3$–manifolds with boundary into the plane, Hiroshima Math. J. 30 (2000) 415–435
• V,A Vassilyev, Lagrange and Legendre characteristic classes, Advanced Studies in Contemporary Mathematics 3, Gordon and Breach, New York (1988)
• T Yamamoto, Classification of singular fibres of stable maps of $4$–manifolds into $3$–manifolds and its applications, J. Math. Soc. Japan 58 (2006) 721–742
• T Yamamoto, Euler number formulas in terms of singular fibers of stable maps, from: “Real and complex singularities”, (A Harris, T Fukui, S Koike, editors), World Sci., Hackensack, NJ (2007) 427–457
• T Yamamoto, Singular fibers of two-colored maps and cobordism invariants, Pacific J. Math. 234 (2008) 379–398