A PREPRINT -J UNE 12, 2020 The Effect of Singularization on the Euler Characteristic M.A. de Jesus Zigart 1 K.A. de Rezende 2 N.G. Grulha Jr. 3 D.V.S. Lima ABSTRACT In this work, singular surfaces are obtained from smooth orientable closed surfaces by applying three basic simple loop operations, collapsing operation, zipping operation and double loop identification, each of which produces different singular surfaces. A formula that provides the Euler characteristic of the singularized surface is proved. Also, we introduce a new definition of genus for singularized surfaces which generalizes the classical definition of genus in the smooth case. A theorem relating the Euler characteristic to the genus of the singularized surface is proved. MSC 2010: 14B05; 14J17; 55N10. 1 Introduction Singularities appear in several fields of study as a sign of qualitative change. We may experience them in Calculus, representing maximum or minimum points of a function; in Dynamical Systems, as stationary solutions that characterize the behaviour of solutions in their vicinity; or in Physics, where they can appear on larger scales, for instance, when a massive star undergoes a gravitational collapse after exhausting its internal nuclear fuel, which can lead to the birth of black holes or naked singularities, the latter being discussed as potential particle accelerators, acting like cosmic super-colliders [7, 8]. The formation of these so called spacetime singularities is a more general phenomena in which general theory of relativity plays an important role [6]. And the most appealing example of such singularity is perhaps the Big Bang. Singularity Theory, structured as a field of research, has risen from the work of Hassler Whitney, John Mather and René Thom. It is the field of science dedicated to studying singularities in their many occurances. One approach is to consider the embedding of a m-dimensional smooth manifold in an Euclidean space of dimension lower than 2m. According to Whitney’s embedding theorem [13], by doing so, the embedding will necessarily cause the manifold to self-intersect, originating singular sets. An interesting question is to consider the types of singularities produced in this fashion and to study their stability. In addition, understanding how smooth manifolds come together at their singular points is another concern in Singularity Theory, studied oftenly under resolutions of singular algebraic surfaces, that is, surfaces given as the zeroset of one polynomial in three variables. Resolving a singular surface means trying to find a surjective map from a smooth surface to the singular surface, which is an isomorphism almost everywhere. Finding blowups, which make up the resolution map, can be highly nontrivial [3]. However, one can perform the inverse process, called a blowdown, to produce singularities on a smooth surface by contracting a hypersurface. It is clear that this construction yields the resolution of the singularity obtained this way. But the singularity produced by an arbitrary blowdown may not be singularity one hopes for. This interplay between blowdowns and blowups can be depicted, for instance, by a continuous deformation of a torus onto a pinched torus. In this scenario, both maps are given intuitively by the initial and final stage of this deformation, as one goes back and forth in time, blowup and blowdown, respectively. The deformation gives rise to a family of surfaces called a smoothing of the singular surface, which in this case is produced by a vanishing cycle. The study of deformations, smoothings, vanishing cycles, unfoldings and bifucartions are oftenly posed, along with resolutions, in order to understand the topology of singular spaces. Brasselet et al., in [2], propose the study by vector field methods, which is very useful to compute the Euler characteristic of such surfaces. For the singular surfaces given as the image of stable mappings, a formula that computes their Euler characteristic was proved in [5] by Izumiya and Marar. 1 Partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001 2 Partially supported by CNPq under grant 305649/2018-3 and FAPESP under grant 2018/13481-0. 3 Supported by CNPq under grant 303046/2016-3; and FAPESP under grant 2017/09620-2. 1 arXiv:2006.06056v1 [math.GT] 10 Jun 2020