Classification of curves on surfaces and free links via homotopy theory of words and phrases

2012 ◽  
Author(s):  
Tomonori Fukunaga
Keyword(s):  
Homotopy Theory ◽  
Curves On Surfaces

scholarly journals Goodwillie Approximations to Higher Categories

2021 ◽  
Vol 272 (1333) ◽  
Author(s):  
Gijs Heuts
Keyword(s):  
Homotopy Theory ◽  
Symmetric Groups ◽  
Homotopy Groups ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Content Type ◽  
Simply Connected ◽  
Derivatives Of ◽  
Connected Spaces

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

On the geometry of homotopy invariants of links

1992 ◽  
Vol 111 (2) ◽  
pp. 291-298 ◽  
Author(s):  
Washington Mio
Keyword(s):  
Knot Theory ◽  
Homotopy Theory ◽  
Geometric Approach ◽  
Surgery Theory ◽  
Codimension Two ◽  
Higher Dimensional ◽  
Link Complements ◽  
Rich Information ◽  
Homotopy Invariants

One of the central problems in higher-dimensional knot theory is the classification of links up to concordance. In 14, Le Dimet constructed a universal model for (disk) link complements, which allowed him to formulate this problem in the framework of surgery theory by applying the Cappell-Shaneson program for studying codimension two embeddings of manifolds 1. The concordance classification was reduced to questions in L-theory (-groups 1) and homotopy theory (of Vogel local spaces 14). While recent results of Cochran and Orr2 (see also 18) provide rich information on the -theoretic part of the problem (in particular, they settle the question of the existence of links not concordant to boundary links), little is known about Le Dimet's homotopy invariant of links; for example, it is not known whether it may ever be non-trivial, or phrasing it more geometrically (according to 19), whether there are links that are not concordant to sublinks of homology boundary links. This motivated us to look at simpler classes of links, for which a more direct geometric approach to the problem is also possible, in an attempt to get some insight on the geometry carried by the homotopy invariants.

scholarly journals HOMOTOPY CLASSIFICATION OF NANOPHRASES IN TURAEV'S THEORY OF WORDS

2009 ◽  
Vol 18 (07) ◽  
pp. 901-915 ◽  
Author(s):  
TOMONORI FUKUNAGA
Keyword(s):  
Crossing Number ◽  
Homotopy Classification ◽  
Two Component ◽  
Curves On Surfaces

The purpose of this paper is to give the homotopy classification of nanophrases of length 2 with 4 letters. To do it we construct some new invariants of nanophrases γ and T. The invariant γ defined in this paper is an extension of the invariant γ for nanowords introduced by Turaev. The invariant T is a new invariant of nanophrases. As a corollary of these results, we give the classification of two-component pointed, ordered, oriented curves on surfaces with minimum crossing number less than or equal to 2.

scholarly journals Pencils on Surfaces with Normal Crossings and the Kodaira Dimension of

2021 ◽  
Vol 9 ◽  
Author(s):  
Daniele Agostini ◽  
Ignacio Barros
Keyword(s):  
Recent Result ◽  
Moduli Space ◽  
Hyperelliptic Curves ◽  
Upper Bounds ◽  
Kodaira Dimension ◽  
Canonical Divisor ◽  
Curves On Surfaces ◽  
Negative Kodaira Dimension

Abstract We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline {\mathcal {M}}_{g,n}$ is not pseudoeffective in some range, implying that $\overline {\mathcal {M}}_{12,6}$ , $\overline {\mathcal {M}}_{12,7}$ , $\overline {\mathcal {M}}_{13,4}$ and $\overline {\mathcal {M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline {\mathcal {M}}_{12,8}$ and $\overline {\mathcal {M}}_{16}$ . We also show that the moduli space of $(4g+5)$ -pointed hyperelliptic curves $\overline {\mathcal {H}}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the classification of moduli of pointed hyperelliptic curves with negative Kodaira dimension.

scholarly journals On homotopy classification of phrases and its application

2010 ◽  
Vol 43 (2) ◽  
Author(s):  
Tomonori Fukunaga
Keyword(s):  
Homotopy Theory ◽  
Homotopy Classification

AbstractThis paper is a survey on the recent study of homotopy theory of generalized phrases in Turaev’s theory of words and phrases. In this paper, we introduce the homotopy classification of generalized phrases with some conditions on numbers of letters. The theory of topology of words and phrases are closely related to the theory of surface-curves. We also introduce applications of topology of words to the topology of surface-curves.

scholarly journals Unique pseudolifting property in digital topology

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 739-746
Author(s):  
Sang-Eon Han
Keyword(s):  
Homotopy Theory ◽  
Digital Topology ◽  
Fundamental Groups ◽  
Digital Version ◽  
Digital Covering ◽  
Digital Spaces

The generalized universal lifting property plays an important role in classical topology. In digital topology we also have its digital version [5, 6, 14]. More precisely, the paper [6] established the concept of a digital covering (see also [11, 17]). It has substantially contributed to the calculation of digital fundamental groups of some digital spaces, the classification of digital spaces and so forth. The paper [6] also established the unique lifting property of a digital covering which plays an important role in studying both digital covering and digital homotopy theory. Motivated by the unique lifting property, the paper develops a pseudocovering which is weaker than a digital covering and investigates its various properties. Furthermore, the paper proves that a pseudocovering with some hypothesis has the unique pseudolifting property which is weaker than the unique lifting property in digital covering theory.

scholarly journals Non-ultra regular digital covering spaces with nontrivial automorphism groups

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1205-1218 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Automorphism Group ◽  
Homotopy Theory ◽  
Automorphism Groups ◽  
Digital Images ◽  
Covering Space ◽  
Transformation Groups ◽  
Covering Spaces ◽  
Nontrivial Automorphism ◽  
Digital Covering

The study of digital covering transformation groups (or automorphism groups, discrete deck transformation groups) plays an important role in the classification of digital spaces (or digital images). In particular, the research into transitive or nontransitive actions of automorphism groups of digital covering spaces is one of the most important issues in digital covering and digital homotopy theory. The paper deals with the problem: Is there a digital covering space which is not ultra regular and has an automorphism group which is not trivial? To solve the problem, let us consider a digital wedge of two simple closed ki-curves with a compatible adjacency, i ?{1,2}, denoted by (X, k). Since the digital wedge (X, k) has both infinite or finite fold digital covering spaces, in the present paper some of these infinite fold digital covering spaces were found not to be ultra regular and further, their automorphism groups are not trivial, which answers the problem posed above. These findings can be substantially used in classifying digital covering spaces and digital images so that the paper improves on the research in Section 4 of [3] (compare Figure 2 of the present paper with Figure 2 of [3]), which corrects an error that appears in the Boxer and Karaca's paper [3] (see the points (0,0), (0,8), (6, -1) and (6,7) in Figure 2 of [3]).

Homotopy of gauge groups over high-dimensional manifolds

2021 ◽  
pp. 1-27
Author(s):  
Ruizhi Huang
Keyword(s):  
Homotopy Theory ◽  
High Dimensional ◽  
Gauge Groups ◽  
Classical Work

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n − 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

On the homomorphisms of sequences

1952 ◽  
Vol 48 (4) ◽  
pp. 521-532 ◽  
Author(s):  
W. H. Cockcroft
Keyword(s):  
Homotopy Theory ◽  
Free Groups ◽  
Algebraic Theory ◽  
Classification Problem ◽  
Two Dimensional ◽  
Homotopy Classification ◽  
Algebraic Classification ◽  
Crossed Modules

The algebraic classification problem with which I shall be concerned in this paper is suggested by the algebraic theory of homomorphisms of free crossed modules whose groups of operators are free groups. This theory arises in the study of the homotopy theory of two-dimensional C.W. complexes*. It is shown in (3) that the homotopy theory of such complexes, including the homotopy classification of mappings, is equivalent to this purely algebraic theory.

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