A topological invariant for stable map germs

1976 ◽  
Vol 32 (2) ◽  
pp. 103-132 ◽  
Author(s):  
James Damon ◽  
Andr� Galligo
Keyword(s):  
Topological Invariant ◽  
Stable Map

scholarly journals The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros

2016 ◽  
Vol 60 (2) ◽  
pp. 319-348 ◽  
Author(s):  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa ◽  
Juan J. Nuño-Ballesteros
Keyword(s):  
Structure Theorem ◽  
Topological Type ◽  
Topological Invariant ◽  
Topological Equivalence ◽  
Stable Maps ◽  
Reeb Graph ◽  
Cone Structure ◽  
Topological Description ◽  
Stable Map

AbstractWe consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

A topological invariant for nonlinear rotations of

Nonlinearity ◽  
1997 ◽  
Vol 10 (1) ◽  
pp. 153-158 ◽  
Author(s):  
Elisabeth Pécou
Keyword(s):  
Topological Invariant

scholarly journals A TORSIONAL TOPOLOGICAL INVARIANT

2007 ◽  
Vol 22 (29) ◽  
pp. 5237-5244 ◽  
Author(s):  
H. T. NIEH
Keyword(s):  
Affine Connection ◽  
Euler Class ◽  
Christoffel Symbol ◽  
Space Time ◽  
Topological Invariant ◽  
Point Of View ◽  
Underlying Space ◽  
Recent Developments ◽  
Theory Point ◽  
Curvature And Torsion

Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

A new invariant and decompositions of manifolds

2018 ◽  
Vol 27 (02) ◽  
pp. 1850019
Author(s):  
Eiji Ogasa
Keyword(s):  
Topological Invariant ◽  
The Body ◽  
Precise Definition ◽  
Closed Manifold ◽  
Compact Manifolds ◽  
Connected Components ◽  
Fixed Base

We introduce a new topological invariant [Formula: see text] of compact manifolds-with-boundaries [Formula: see text] which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let [Formula: see text] and [Formula: see text] be [Formula: see text]-dimensional compact manifolds-with-boundaries. Let [Formula: see text] be a boundary-union of [Formula: see text] and [Formula: see text]. Then we have [Formula: see text] We define [Formula: see text] as follows: First, define an invariant of [Formula: see text]-closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base [Formula: see text], where we impose the condition that the base [Formula: see text] is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base [Formula: see text]. It is our another invariant [Formula: see text]. Take the maximum of the minimum, [Formula: see text], for all basis to satisfy the above condition. It is [Formula: see text]. See the body of the paper for the precise definition.

scholarly journals Generalizations of the Nieh-Yan topological invariant

2021 ◽  
Vol 104 (8) ◽  
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez
Keyword(s):  
Topological Invariant

THE DENEF–LOESER ZETA FUNCTION IS NOT A TOPOLOGICAL INVARIANT

2002 ◽  
Vol 65 (01) ◽  
pp. 45-54 ◽  
Author(s):  
E. ARTAL BARTOLO ◽  
P. CASSOU-NOGUÈS ◽  
I. LUENGO ◽  
A. MELLE HERNÁNDEZ
Keyword(s):  
Zeta Function ◽  
Topological Invariant
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