The general quadruple point formula (original) (raw)

On the triple points of singular maps

Commentarii Mathematici Helvetici, 2002

The number of triple points (mod 2) of a self-transverse immersion of a closed 2nmanifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9]. If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer-and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds.

Quadratic functions in geometry, topology,and M-theory

2002

We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension 4k+24k+24k+2 in terms of a holomorphic line bundle on the abelian variety H2k+1(M)otimesR/ZH^{2k+1}(M)\otimes R/ZH2k+1(M)otimesR/Z. Our results are inspired by work of Witten on the fivebrane partition function in MMM-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of "differential functions." These differential functions generalize the differential characters of Cheeger-Simons, and the bulk of this paper is devoted to their study.

Generalized Monodromy Conjecture in dimension two

Geometry & Topology, 2012

The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ f : (X, 0) → (C, 0) defined on a normal surface singularity (X, 0). The article targets the 'right' extension in the case when the link of (X, 0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f, ω; s) for any f and analytic differential form ω, which will play the key technical localization tool in the later definitions and proofs.

The Euler-Poincaré Characteristic and Mixed Multiplicities

Kyushu Journal of Mathematics, 2015

This paper defines mixed multiplicity systems; the Euler-Poincare characteristic and the mixed multiplicity symbol of N d-graded modules with respect to a mixed multiplicity system, and proves that the Euler-Poincare characteristic and the mixed multiplicity symbol of any mixed multiplicity system of the type (k1,. .. , k d) and the (k1,. .. , k d)-difference of the Hilbert polynomial are the same. As an application, we get results for mixed multiplicities.

COHOMOLOGY JUMP LOCI OF 3-MANIFOLDS

Manuscripta Mathematica , 2022

The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems , and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

Multiplicity as an Object of Study in Contemporary Differential Geometry

ANNALS OF THE ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering., 2013

The study presents introductory concepts of multiplicity as one of the basic concepts of differential topology. The work gives the definition of Hausdorff space and some theorems concerning Hausdorff space.

Minicourse: Cohomology jump loci in geometry and topology. Part IV: Applications

The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.

The double point formula with isolated singularities and canonical embeddings

Journal of the London Mathematical Society, 2020

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus pg = 5: the canonical model is embedded in P 4 if and only if we have a complete intersection of type (2, 4) or (3, 3). Contents 1. Introduction 2 2. The classical double point formula 6 3. Extension of Severi's Statement 9 4. Comparing normal sheaf and quotient bundle 12 5. Surfaces with rational double points 14 6. Double point formulae via symplectic approximations. 20 References 21

MULTIPLE-POINT FORMULAS—A NEW POINT OF VIEW

2002

On the basis of the Generalized Pontryagin-Thom construction (see Rimanyi & Szucs, 1998) and its application in computing Thom polynomials (see Rimanyi, 2001) here we introduce a new point of view in multiple-point theory. Using this approach we will first show how to reprove results of Kleiman and his followers (the corank 1 case) then we will prove some new multiple-point formulas which are not subject to the condition of corank≤ 1.