On the topology of real analytic maps

2014 ◽  
Vol 25 (07) ◽  
pp. 1450069 ◽  
Author(s):  
José Luis Cisneros-Molina ◽  
José Seade ◽  
Nivaldo G. Grulha
Keyword(s):  
Critical Point ◽  
Euler Characteristic ◽  
Euler Characteristics ◽  
Critical Set ◽  
Gauss Theorem ◽  
Real Analytic ◽  
The Difference ◽  
Analytic Maps ◽  
Zero Locus ◽  
Gradient Vector Field

We study the topology of the fibers of real analytic maps ℝn → ℝp, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : ℝn → ℝp, p ≥ 1, with arbitrary critical set, has a Milnor–Lê type fibration away from the discriminant. Now assume also that f has the Thom af-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : ℝn → ℝk with an isolated critical point at the origin. We have Milnor–Lê type fibrations for f and for (f, g) : ℝn → ℝp+k, and we prove for these the analogous of the classical Lê–Greuel formula, expressing the difference of the Euler characteristics of the fibers Ff and Ff,g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map [Formula: see text] on Ff associated to g; as the number of critical points of [Formula: see text] on Ff; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers Ff and Ff,g, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map.

scholarly journals Higher-order orbifold Euler characteristics for compact Lie group actions

2015 ◽  
Vol 145 (6) ◽  
pp. 1215-1222 ◽  
Author(s):  
S. M. Gusein-Zade ◽  
I. Luengo ◽  
A. Melle-Hernández
Keyword(s):  
Euler Characteristic ◽  
Lie Group ◽  
Group Actions ◽  
Wreath Products ◽  
Higher Order ◽  
Compact Lie Group ◽  
Euler Characteristics ◽  
Finite Group Actions ◽  
Lie Group Actions ◽  
Generating Series

We generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.

scholarly journals On the Lê-Milnor fibration for real analytic maps

2016 ◽  
Vol 290 (2-3) ◽  
pp. 382-392 ◽  
Author(s):  
Aurélio Menegon Neto ◽  
José Seade
Keyword(s):  
Milnor Fibration ◽  
Real Analytic ◽  
Analytic Maps

scholarly journals Euler characteristics and their congruences in the positive rank setting

2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Selmer Groups ◽  
Euler Characteristics ◽  
Rank Zero ◽  
Homology Groups ◽  
Main Conjecture ◽  
Congruence Relations ◽  
Higher Rank ◽  
Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

2000 ◽  
Vol 6 (3) ◽  
pp. 311-330 ◽  
Author(s):  
Jan Krajíček ◽  
Thomas Scanlon
Keyword(s):  
Euler Characteristic ◽  
Order Structure ◽  
Euler Characteristics ◽  
Grothendieck Ring ◽  
Open Problems ◽  
Partially Ordered ◽  
Counting Functions ◽  
Finite Structures ◽  
Definable Sets ◽  
Grothendieck Rings

AbstractWe recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

scholarly journals MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES

2019 ◽  
Vol 236 ◽  
pp. 251-310 ◽  
Author(s):  
MARC LEVINE
Keyword(s):  
Euler Characteristic ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Vector Bundles ◽  
Reductive Group ◽  
Characteristic Classes ◽  
Euler Characteristics ◽  
Symmetric Powers ◽  
Maximal Tori ◽  
General Calculus

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

scholarly journals Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1244 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Fixed Point Theory ◽  
Geometric Realization ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Euler Characteristics ◽  
Surface Theory ◽  
Closed Surfaces

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x ∈ S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 ′ do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al.

scholarly journals A Riemann–Hurwitz Theorem for the Algebraic Euler Characteristic

2017 ◽  
Vol 60 (3) ◽  
pp. 490-509
Author(s):  
Andrew Fiori
Keyword(s):  
Euler Characteristic ◽  
Euler Characteristics ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Irreducible Components ◽  
Arbitrary Dimensions ◽  
Smooth Projective Varieties ◽  
Ramification Locus ◽  
Hurwitz Theorem

AbstractWe prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

scholarly journals The computation of the index of a Morse function at a critical point

1988 ◽  
Vol 11 (4) ◽  
pp. 721-726 ◽  
Author(s):  
Takis Sakkalis
Keyword(s):  
Critical Point ◽  
Linear Function ◽  
Euler Characteristic ◽  
Theoretical Approach ◽  
Morse Function ◽  
Singular Hypersurface

A theoretical approach in computing the index of a Morse function at a critical point on a real non-singular hypersurfaceVis given. As a consequence the Euler characteristic ofVis computed. In the case where the hypersurface is polynomial and compact, a procedure is given that finds a linear functionℓ, whose restrictionℓ|V, is a Morse function onV.

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