اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe Digital Hopf Construction
72
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and far between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration which mimics the smooth setting.
قيم البحث
اقرأ أيضاً
Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the mo
st basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.
With a view towards providing tools for analyzing and understanding digitized images, various notions from algebraic topology have been introduced into the setting of digital topology. In the ordinary topological setting, invariants such as the funda
mental group are invariants of homotopy type. In the digital setting, however, the usual notion of homotopy leads to a very rigid invariance that does not correspond well with the topological notion of homotopy invariance. In this paper, we establish fundamental results about subdivision of maps of digital images with $1$- or $2$-dimensional domains. Our results lay the groundwork for showing that the digital fundamental group is an invariant of a much less rigid equivalence relation on digital images, that is more akin to the topological notion of homotopy invariance. Our results also lay the groundwork for defining other invariants of digital images in a way that makes them invariants of this less rigid equivalence.
We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously establish
In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a
graph. The clique complex is a simplicial complex and its edge group is well-known to be isomorphic to the ordinary (topological) fundamental group of its geometric realization. This identification of our intrinsic digital fundamental group with a topological fundamental group---extrinsic to the digital setting---means that many familiar facts about the ordinary fundamental group may be translated into their counterparts for the digital fundamental group: The digital fundamental group of any digital circle is $mathbb{Z}$; a version of the Seifert-van Kampen Theorem holds for our digital fundamental group; every finitely presented group occurs as the (digital) fundamental group of some digital image. We also show that the (digital) fundamental group of every 2D digital image is a free group.
The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category A. It is kn
own that the category of bicommutative (i.e. commutative and cocommutative) Hopf algebras over a field k is an abelian category. Denote the category by H. In this paper, we give some ways to construct H-valued homology theories. As a main result, we give H-valued homology theories whose coefficients are neither group Hopf algebras nor function Hopf algebras. The examples contain not only ordinary homology theories but also extraordinary ones.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
