First-order local invariants of stable maps from 3-manifolds to \(\mathbb R^{3}\). (English) Zbl 1266.57018
The authors study Vassiliev type invariants of stable maps from 3-manifolds to \(\mathbb R^3\). These invariants are locally constant functions on the space of stable maps. The idea of Vassiliev invariants is to relate these functions with the topology of the discriminant set. The authors describe the codimension 1 and codimension 2 degenerations of (multi-) germs from 3-dimensions to 3-dimensions, and hence obtain a complete set of generators for first order Vassiliev invariants. Two of these invariants are related to monosingularities, one to a multisingularity, and one to the Euler charactersitic of a branch set. The authors also describe interesting relations to first order Vassiliev invariants of maps from 2-dimension to 3-dimension (found by Goryunov).
Reviewer: Richárd Rimányi (Chapel Hill)
MSC:
| 57R45 | Singularities of differentiable mappings in differential topology |
| 58K65 | Topological invariants on manifolds |
References:
| [1] | V. I. Arnold, Topological invariants of plane curves and caustics, Univ. Lecture Ser., 5, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0858.57001 |
| [2] | —, Plane curves, their invariants, perestroikas and classifications, Adv. Soviet Math., 21, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0864.57027 |
| [3] | J. W. Bruce, A classification of 1-parameter families of map germs \(R^{3},0\to R^{3},0\) with applications to condensation problems, J. London Math. Soc. (2) 33 (1986), 375-384. · Zbl 0556.58006 · doi:10.1112/jlms/s2-33.2.375 |
| [4] | T. Cooper, D. Mond, and R. Wik Atique, Vanishing topology of codimension 1 multi-germs over \(R\) and \(C,\) Compositio Math. 131 (2002), 121-160. · Zbl 1002.32016 · doi:10.1023/A:1014930205374 |
| [5] | M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math., 14, Springer-Verlag, New York, 1973. · Zbl 0294.58004 |
| [6] | V. Goryunov, Singularities of projections of complete intersections, J. Soviet Math. 27 (1984), 2785-2811. · Zbl 0555.58009 |
| [7] | —, Local invariants of mappings of surfaces into three space, The Arnold-Gelfand mathematical seminars, pp. 223-225, Birkhäuser, Boston, 1997. · Zbl 0887.58007 · doi:10.1007/978-1-4612-4122-5_11 |
| [8] | D. Hacon, C. Mendes de Jesus, and M. C. Romero Fuster, Topological invariants of stable maps from a surface to the plane from a global viewpoint, Real and complex singularities, 227-235, Lecture Notes in Pure and Appl. Math., 232, Dekker, New York, 2003. · Zbl 1079.58512 |
| [9] | —, Fold maps from the sphere to the plane, Experiment. Math. 15 (2006), 491-497. · Zbl 1133.57019 · doi:10.1080/10586458.2006.10128973 |
| [10] | —, Stable maps from surfaces to the plane with prescribed branching data, Topology Appl. 154 (2007), 166-175. · Zbl 1107.57018 · doi:10.1016/j.topol.2006.04.005 |
| [11] | K. Houston, Black and white local invariants of mappings from surfaces into three-space, Period. Math. Hungar. 49 (2004), 49-72. · Zbl 1075.57011 · doi:10.1023/B:MAHU.0000040539.55454.9e |
| [12] | S. Izumiya and W. L. Marar, On topologically stable singular surfaces in a 3-manifold, J. Geom. 52 (1995), 108-119. · Zbl 0840.57010 · doi:10.1007/BF01406831 |
| [13] | W. L. Marar and F. Tari, On the geometry of simple germs of co-rank 1 maps from \(R^{3}\) to \(R^{3},\) Math. Proc. Cambridge Philos. Soc. 119 (1996), 469-481. · Zbl 1005.57501 · doi:10.1017/S030500410007434X |
| [14] | J. A. Montaldi, On contact between submanifolds, Michigan Math. J. 33 (1986), 195-199. · Zbl 0601.53007 · doi:10.1307/mmj/1029003348 |
| [15] | A. M. Montesinos Amilibia, Superficies II, computer program available by anonymous ftp at http://www.uv.es/\~,montesin/. |
| [16] | T. Ohmoto and F. Aicardi, First order local invariants of apparent contours, Topology 45 (2006), 27-45. · Zbl 1082.57024 · doi:10.1016/j.top.2005.04.005 |
| [17] | R. Oset Sinha, Topological invariants of stable maps from 3-manifolds to three-space, Ph.D. thesis, Universitat de València, 2009. · Zbl 1207.58029 |
| [18] | V. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, pp. 23-69, Adv. Soviet Math., 1, Amer. Math. Soc., Providence, RI, 1990. |
| [19] | —, Complements of discriminants of smooth maps: Topology and applications, Transl. Math. Monogr., 98, Amer. Math. Soc., Providence, RI, 1992. |
| [20] | M. Yamamoto, First order semi-local invariants of stable maps of 3-manifolds into the plane, Proc. London Math. Soc. (3) 92 (2006), 471-504. · Zbl 1092.57020 · doi:10.1112/S0024611505015534 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
