Geometriae Dedicata

, Volume 194, Issue 1, pp 187–207 | Cite as

Topological invariants of stable maps of oriented 3-manifolds in \(\mathbb {R}^4\)

  • C. CasonattoEmail author
  • M. C. Romero Fuster
  • R. G. Wik Atique
Original Paper


We study the \(\mathbb {Z}\)-module of first order local Vassiliev type invariants of stable maps of oriented 3-manifolds into \(\mathbb {R}^4\). As a previous step, we determine a complete classification of the codimension two germs and multigerms as well as their corresponding bifurcation diagrams. This allows us to show the existence of 4 generators for this module. In particular, we see that the number of pairs of quadruple point, the number of transversal intersections of the crosscap suspension with immersive branches and the Euler number of the image are first order local Vassiliev type invariants for these maps. We also prove that the total number of connected components of the triple point curve is a non local Vassiliev type invariant.


Singularities Stable maps First order Vassiliev invariants 

Mathematics Subject Classification (2000)

57R45 58K15 58K65 



Funding was provided by DGCYT and FEDER (Grant No. MTM2015-64013-P).


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal de Uberlândia-MGUberlândiaBrazil
  2. 2.Departamento de Geometria i Topologia, Facultat de MatemàtiquesUniversitat de ValènciaBurjassot, ValenciaSpain
  3. 3.Departamento de MatemáticaICMC/USP-São CarlosSão CarlosBrazil

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