scholarly journals On Homotopy Invariants of Combings of Three-manifolds

2015 ◽  
Vol 67 (1) ◽  
pp. 152-183 ◽  
Author(s):  
Christine Lescop
Keyword(s):  
Vector Fields ◽  
Euler Class ◽  
Integral Invariant ◽  
Alternative Definition ◽  
Rational Homology ◽  
Homotopy Classes ◽  
Torsion Element ◽  
3 Dimensional ◽  
Linking Number ◽  
Homotopy Invariants

AbstractCombings of compact, oriented, 3-dimensional manifoldsMare homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying Spinc-structure. A combing is called torsion if this Euler class is a torsion element of H2(M; Z). Gompf introduced a Q-valued invariant θGof torsion combings on closed 3-manifolds, and he showed that θGdistinguishes all torsion combings with the same Spinc-structure. We give an alternative definition for θGand we express its variation as a linking number. We define a similar invariantp1of combings for manifolds bounded by S2. We relate p1 to the Θ-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula Θ = ¼P1+ 6λ, where λ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

Euler classes of vector bundles over manifolds

2021 ◽  
Vol 71 (1) ◽  
pp. 199-210
Author(s):  
Aniruddha C. Naolekar
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere ◽  
Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

Immersions with non-zero normal vector fields

1992 ◽  
Vol 112 (2) ◽  
pp. 281-285 ◽  
Author(s):  
Bang-He Li ◽  
Gui-Song Li
Keyword(s):  
Vector Fields ◽  
Normal Vector ◽  
Homotopy Classes ◽  
Natural Action ◽  
Singularity Method ◽  
One To One ◽  
Regular Homotopy

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

scholarly journals New Formulas and Results for 3-Dimensional Vector Fields

2021 ◽  
Vol 12 (11) ◽  
pp. 1058-1096
Author(s):  
Sadanand D. Agashe
Keyword(s):  
Vector Fields ◽  
Dimensional Vector ◽  
3 Dimensional

Suspension of the Lusternik-Schnirelmann Category

1970 ◽  
Vol 22 (6) ◽  
pp. 1129-1132
Author(s):  
William J. Gilbert
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Category Structure ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Cw Complexes ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected ◽  
Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

A COMBINATORIAL APPROACH TO LINKING NUMBERS IN RATIONAL HOMOLOGY SPHERES

2000 ◽  
Vol 09 (05) ◽  
pp. 703-711
Author(s):  
CARL J. STITZ
Keyword(s):  
Combinatorial Approach ◽  
Rational Homology ◽  
Characteristic Curves ◽  
Combinatorial Description ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Linking Numbers ◽  
Heegaard Diagram ◽  
Rational Homology Spheres ◽  
Matrix Invariants

In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.

scholarly journals Vector Fields and Genus in Dimension 3

2020 ◽  
Author(s):  
Pierre Dehornoy ◽  
Ana Rechtman
Keyword(s):  
Vector Field ◽  
Invariant Measure ◽  
Euler Characteristic ◽  
Boundary Data ◽  
Vector Fields ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere

Abstract Given a vector field on a 3D rational homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. This provides a formula for the genus of a transverse surface, and in particular, of a Birkhoff section. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to helicity.

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