Displaying similar documents to “Homology with equationally compact coefficients”

Relationship among various Vietoris-type and microsimplicial homology theories

Takuma Imamura (2021)

Archivum Mathematicum

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In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology...

Steenrod homology

Yu. T. Lisitsa, S. Mardešić (1986)

Banach Center Publications

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Homology of representable sets

Marian Mrozek, Bogdan Batko (2010)

Annales Polonici Mathematici

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We generalize the notion of cubical homology to the class of locally compact representable sets in order to propose a new convenient method of reducing the complexity of a set while computing its homology.

On the first homology of Peano continua

Gregory R. Conner, Samuel M. Corson (2016)

Fundamenta Mathematicae

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We show that the first homology group of a locally connected compact metric space is either uncountable or finitely generated. This is related to Shelah's well-known result (1988) which shows that the fundamental group of such a space satisfies a similar condition. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result does not follow from Shelah's. We clarify a claim made by Pawlikowski (1998) and offer...

A 2-category of chronological cobordisms and odd Khovanov homology

Krzysztof K. Putyra (2014)

Banach Center Publications

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We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex whose homotopy type modulo certain relations is a link invariant. Both the original and...

Transverse Homology Groups

S. Dragotti, G. Magro, L. Parlato (2006)

Bollettino dell'Unione Matematica Italiana

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We give, here, a geometric treatment of intersection homology theory.

A computation in Khovanov-Rozansky homology

Daniel Krasner (2009)

Fundamenta Mathematicae

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We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl₃-homology.

On total homology

L. D. Mdzinarishvili (1986)

Banach Center Publications

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Chewing the Khovanov homology of tangles

Magnus Jacobsson (2004)

Fundamenta Mathematicae

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We present an elementary description of Khovanov's homology of tangles [K2], in the spirit of Viro's paper [V]. The formulation here is over the polynomial ring ℤ[c], unlike [K2] where the theory was presented over the integers only.

Torsion in one-term distributive homology

Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra (2014)

Fundamenta Mathematicae

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The one-term distributive homology was introduced in [Prz] as an atomic replacement of rack and quandle homology, which was first introduced and developed by Fenn-Rourke-Sanderson [FRS] and Carter-Kamada-Saito [CKS]. This homology was initially suspected to be torsion-free [Prz], but we show in this paper that the one-term homology of a finite spindle may have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their...

Khovanov homology, its definitions and ramifications

Oleg Viro (2004)

Fundamenta Mathematicae

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Mikhail Khovanov defined, for a diagram of an oriented classical link, a collection of groups labelled by pairs of integers. These groups were constructed as the homology groups of certain chain complexes. The Euler characteristics of these complexes are the coefficients of the Jones polynomial of the link. The original construction is overloaded with algebraic details. Most of the specialists use adaptations of it stripped off the details. The goal of this paper is to overview these...