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The use of Cauchy's method in proving the well-known Euler formula is an object of many controversies. The purpose of this paper is to prove that the Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus., Comment: in Portuguese

The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus.

We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$. The aim of the paper is to provide partial answers to the following questions: Does the bi-Lipschitz type of a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ determine the bi-Lipschitz type of the link of $f$ and of the double point set of $f$? Reciprocally, given a map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$, do the bi-Lipschitz types of the link of $f$ and of the double point set of $f$ determine the bi-Lipschitz type of the germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$? We provide a positive answer to the first question in the case of a finitely determined map germ $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)$ where $2n-1 \leq p$ (Theorem 3.3). With regard to the second question, for a finitely determined map germ $f : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0),$ we show that a complete set of invariants for the bi-Lipschitz classification with respect to the inner metric of $X_f=f(U)$, where $U$ is a small neighbourhood of the origin in $\mathbb R^2$, is is given by the link of $f$, the image of the double point set of $f$ and the polar curve of a generic projection into the plane (Proposition 4.13). In particular, in the homogeneous parametrization case $f: (\mathbb{R}^2, 0) \to (\mathbb{R}^3, 0)$ of corank 1, we do not need the hypothesis on the equivalence of the image of the double point set (Theorem 5.2). Finally, we apply our results to relate the $C^{0}- \mathcal A$ classes of finitely determined map germs $f$ of corank 1 with homogeneous parametrization and the inner bi-Lipschitz type of $X_f$ (Proposition 5.4).

The Poincare-Hopf Theorem is one of the most used in other areas of science. There are applications of the Poincare-Hopf Theorem in physics, chemistry, biology and even in economics, psychology, etc ... The Poincare-Hopf Theorem connects an invariant of combinatorial, the character of Euler-Poincare to an invariant of differential geometry, index of vector fields. The results that connect two very different areas of mathematics can be considered as the most beautiful, useful and fruitful., Comment: in Portuguese

For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi- Bracci-Izawa-Suwa (2016) in the framework of Cech-de Rham cohomology. We take up the problem from the combinatorial viewpoint and give some finer results, in particular for the local classes. As to the global class, we clarify the relation with the cohomology coincidence class as studied by Biasi-Libardi-Monis (2015). In fact they introduced such a class in the context of several maps and we also consider this case. In particular we define the local homology class and give some explicit expressions. These all together lead to a generalization of the classical Lefschetz coincidence point formula., Comment: 18 pages

We introduce the Euler-Poincar\'e's characteristic with an elementary way and historically. We explain also why one should call Descartes-Poincar\'e characteristic instead of the Euler-Poincar\'e's characteristic. All the considered spaces are compact and without boundary. We work essentially on smooth and oriented surfaces . However, we work also on singular and non-oriented surfaces. We provide also the results which may be extended for the case of superior dimension., Comment: in Portuguese

In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this context ., Comment: 22 pages, To appear in Advances in Mathematics

In this paper we give a formula for the Hirzebruch $\chi_y$-genus $\chi_y(X)$ and similarly for the motivic Hirzebruch class $T_{y*}(X)$ for possibly singular varieties $X$, using the Vandermonde matrix. Motivated by the notion of secondary Euler characteristic and higher Euler characteristic, we consider a similar notion for the motivic Hirzebruch class, which we call a \emph{derived motivic Hirzebruch class}, Comment: 19 pages, comments are welcome: we have made some revisions; to appear in Homology, Homotopy and Applications

We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We prove in dimension $3$ a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing previous results by J. Snoussi in dimension $2$. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of L\^e D. T. concerning constancy of the Milnor number.

In this article we study the bordism groups of normally nonsingular maps $f: X \to Y$ defined on pseudomanifolds $X$ and $Y$. To characterize the bordism of such maps, inspired by the formula given by Stong, we give a general definition of Stiefel-Whitney numbers defined on $X$ and $Y$ using the Wu classes defined by Goresky and Pardon and we show that in several cases the cobordism class of a normally nonsingular map $f:X \to Y$ guarantees that these numbers are zero., Comment: 11 pages

Let $X$ be a real-analytic manifold and $g\colon X\to{\mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $\omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $\omega$ has a relative primitive: there is a subanalytic $(r-1)$-form $\Omega$ such that $dg\Lambda (\omega-d\Omega)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of "relative Whitney forms" associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of Fran\c{c}ois Treves.

We give a short proof of the fact that the Chern classes for singular varieties defined by Marie-Helene Schwartz by means of "radial frames" agree with the functorial notion defined by Robert MacPherson., Comment: 6 pages

In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total \lambda-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation T_{y*}: K_{0}(var/X)-> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. T_{y*} is a homology class version of the motivic measure corresponding to suitable specialization of the well known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y=-1), the Todd class transformation of the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for y=0) and the L-class transformation of Cappell-Shaneson (for y=1). In the simplest case of a normal Gorenstein variety with ``canonical singularities'' we also explain a relation among the ``stringy version'' of our characteristic classses, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe. Moreover, all our results can be extended to varieties over a base field k of characteristic 0., Comment: 55 pages, some typos corrected, this is a completely new and improved version of the paper math.AG/0405412

In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y=-1) and the Todd class transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for y=0). In fact, T_{y} is the composition of a generalized version of this Todd class transformation due to Yokura, and a new motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total lambda-class of the cotangent bundle to singular spaces. Here G_{0}(X) is the Grothendieck group of coherent sheaves, and the construction of mC_{*} is based on some results from the theory of algebraic mixed Hodge modules due to M.Saito. In the final part of the paper we use the algebraic cobordism theory of Levine and Morel to lift mC_{*} further up to a natural transformation mC'_{*} from K_{0}(var/X) to a suitable universal Borel-Moore weak homology theory. Moreover, all our results can be extended to varieties over a base field k, which can be embedded into the complex numbers., Comment: 32 pages, no figures, some misprints and list of authors corrected, references updated

The existence of bivariant Chern classes was conjectured by W.Fulton and R.MacPherson and proved by J.P.Brasselet for cellular morphisms of analytic varieties. In this paper we show that restricted to morphisms whose target varieties are possible singular but (rational) homology manifolds, the bivariant Chern classes (with rational coefficients) are uniquely determined. In the final sections we construct a unique bivariant Chern class satisfying a suitable normalization condition. In fact, it will be a special case of a general construction of unique Grothendieck transformations, which in a sense gives a positive answer to a corresponding question posed by Fulton and MacPherson. This construction generalizes and unifies previous attemps, in particular the main result of the paper math.AG/0209299., Comment: 24 pages, no figures

The proof of the combinatorial Hard Lefschetz Theorem for the ``virtual'' intersection cohomology of a not necessarily rational polytopal fan that has been presented by K. Karu completely establishes Stanley's conjectures for the generalized $h$-vector of an arbitrary polytope. The main ingredients, namely, Poincare Duality and the Hard Lefschetz Theorem, both rely on the intersection product. In the constructions of Barthel, Brasselet, Fieseler and Kaup and Bressler and Lunts, there remained an apparent ambiguity. The recent solution of this problem by Bressler and Lunts uses the formalism of derived categories. The present article gives a straightforward approach to combinatorial duality and a natural intersection product, completely within the framework of elementary sheaf theory and commutative algebra, thus avoiding derived categories., Comment: Plain TeX, 21 pages

We show that the Chern-Schwartz-MacPherson class of a hypersurface X in a nonsingular variety M `interpolates' between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern-Schwartz-MacPherson class of such `nice' hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern-Schwartz-MacPherson class of X in terms of its polar classes., Comment: 10 pages

We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These sheaves are models for the "pure" objects of a "perverse category"; a "Decomposition Theorem" is shown to hold. -- Formalizing those fans that define "equivariantly formal" toric varieties (where equivariant and non-equivariant intersection cohomology determine each other by Kunneth type formulae), we study "quasi-convex" fans (including fans with convex or with "co-convex" support). For these, there is a meaningful "virtual intersection cohomology". We characterize quasi-convex fans by a topological condition on the support of their boundary fan and prove a generalization of Stanley's "Local-Global" formula realizing the intersection Poincare polynomial of a complete toric variety in terms of local data. Virtual intersection cohomology of quasi-convex fans is shown to satify Poincare duality. To describe the local data in terms of virtual intersection cohomology of lower-dimensional complete polytopal fans, one needs a "Hard Lefschetz" type theorem. It requires a vanishing condition that is known to hold for rational cones, but yet remains to be proven in the general case., Comment: 41 pages, Plain TeX ("private" macros included), submitted for publication

We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second of these axioms, a key role is played by "equivariantly formal" toric varieties, where equivariant and "usual" (non-equivariant) intersection cohomology determine each other by Kunneth type formulae. Minimal extension sheaves can be constructed in a purely formal way and thus also exist for non-rational fans. As a consequence, we can extend the notion of an equivariantly formal fan even to this general setup. In this way, it will be possible to introduce "virtual" intersection cohomology for equivariantly formal non-rational fans., Comment: 31 pages, AMS-Latex (all "private" macros included), to be published in "Algebraic Geometry - Hirzebruch 70" (Proceedings of the conference at the Banach Centre, Warszawa, May 1998), Contemporary Mathematics, AMS

There are several ways to generalize characteristic classes for singular algebraic varieties. The simplest ones to describe are Chern-Mather classes obtained by Nash blow up. They serve as an ingredient to construct Chern-MacPherson-Schwartz classes. Unfortunately, they all are defined in homology. There are examples showing, that they do not lie in the image of Poincar\'e morphism. On the other hand they are represented by an algebraic cycles. Barthel, Brasselet, Fiesler, Kaup and Gabber have shown that, any algebraic cycle can be lifted to intersection homology. Nevertheless, a lift is not unique. The Chern-Mather classes are represented by polar varieties. We show how to define a canonical lift of Chern-Mather classes to intersection homology. Instead of the polar variety alone, we consider it as a term in the whole sequence of inclusions of polar varieties. The inclusions are of codimension one. In this case the lifts are unique., Comment: 17 pages, plain TeX

A classical result of A. Connes asserts that the Frechet algebra of smooth functions on a smooth compact manifold X provides, by a purely algebraic procedure, the de Rham cohomology of X. Namely the procedure uses Hochschild and cyclic homology of this algebra. In the situation of a Thom-Mather stratified variety, we construct a Frechet algebra of functions on the regular part and a module of poles along the singular part. We associate to these objects a complex of differential forms and an Hochschild complex, on the regular part, both with poles along the singular part. The de Rham cohomology of the first complex and the cylic homology of the second one are related to the intersection homology of the variety, the corresponding perversity is determined by the orders of poles., Comment: TeX, 16 pages