scholarly journals Extended quandle spaces and shadow homotopy invariants of classical links

2017 ◽  
Vol 26 (03) ◽  
pp. 1741010
Author(s):  
Seung Yeop Yang
Keyword(s):  
Classical Link ◽  
Homotopy Invariant ◽  
Homotopy Invariants

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.

scholarly journals FINITE TYPE LINK HOMOTOPY INVARIANTS II: Milnor's $\bar\mu$-Invariants

2000 ◽  
Vol 09 (06) ◽  
pp. 735-758 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Higher Order ◽  
Finite Type Invariants ◽  
Homotopy Invariant ◽  
Type Invariant ◽  
Link Homotopy ◽  
Homotopy Invariants

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's link homotopy invariant [Formula: see text] is a finite type invariant, of type 1, in this sense. We also generalize this approach to Milnor's higher order [Formula: see text] invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

scholarly journals Smooth functorial field theories from B-fields and D-branes

2021 ◽  
Vol 16 (1) ◽  
pp. 75-153
Author(s):  
Severin Bunk ◽  
Konrad Waldorf
Keyword(s):  
Field Theory ◽  
Sigma Models ◽  
Lagrangian Approach ◽  
Field Theories ◽  
Open Strings ◽  
Homotopy Invariant ◽  
Target Manifold ◽  
Definition Of ◽  
Smooth Map

AbstractIn the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.

scholarly journals A motivic version of the theorem of Fontaine and Wintenberger

2018 ◽  
Vol 155 (1) ◽  
pp. 38-88 ◽  
Author(s):  
Alberto Vezzani
Keyword(s):  
Galois Groups ◽  
Rational Homotopy ◽  
Mixed Characteristic ◽  
Homotopy Invariant ◽  
Analytic Varieties

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat }$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^{\flat }$ are isomorphic.

Spectra of derived module homomorphisms

1987 ◽  
Vol 101 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Alan Robinson
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Theory ◽  
Ring Spectrum ◽  
Homotopy Invariant ◽  
New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

Homotopy invariants and continuous mappings

1950 ◽  
Vol 202 (1069) ◽  
pp. 253-263 ◽  
Keyword(s):  
Homotopy Type ◽  
Betti Numbers ◽  
Continuous Mappings ◽  
Numerical Invariants ◽  
Homotopy Invariants

This paper contributes new numerical invariants to the topology of a certain class of polyhedra. These invariants, together with the Betti numbers and coefficients of torsion, characterize the homotopy type of one of these polyhedra. They are also applied to the classification of continuous mappings of an ( n + 2)-dimensional polyhedron into an ( n + 1)-sphere ( n > 2).

Isotopy invariants of topological spaces

1960 ◽  
Vol 255 (1282) ◽  
pp. 331-366 ◽  
Keyword(s):  
Homotopy Type ◽  
Algebraic Topology ◽  
Complete System ◽  
Topological Spaces ◽  
Recent Developments ◽  
The Difference ◽  
Linear Graphs ◽  
Homotopy Invariants ◽  
General Method

A few decades ago, topologists had already emphasized the difference between homotopy and isotopy. However, recent developments in algebraic topology are almost exclusively on the side of homotopy. Since a complete system of homotopy invariants has been obtained by Postnikov, it seems that hereafter we should pay more attention to isotopy invariants and new efforts should be made to attack the classical problems. The purpose of this paper is to introduce and study new algebraic isotopy invariants of spaces. A general method of constructing these invariants is given by means of a class of functors called isotopy functors. Special isotopy functors are constructed in this paper, namely, the m th residual functor R m and the m th. enveloping functor E m . Applications of these isotopy invariants to linear graphs are given in the last two sections. It turns out that these invariants can distinguish various spaces belonging to the same homotopy type.

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