Main properties of singular homology groups and their computation

2001 ◽  
pp. 83-94
Author(s):  
V. Vassiliev
Keyword(s):  
Homology Groups ◽  
Singular Homology

scholarly journals Elementary Quotients of Abelian Groups, and Singular Homology on Manifolds

1970 ◽  
Vol 39 ◽  
pp. 167-198 ◽  
Author(s):  
Marston Morse ◽  
Stewart Scott Cairns
Keyword(s):  
Abelian Groups ◽  
Betti Numbers ◽  
Topological Manifold ◽  
Homology Groups ◽  
Singular Homology

Let there be given a compact topological manifold Mn. If Mn admits a “triangulation” it is known that the fundamental invariants, namely the connectivities of Mn over fields, the Betti numbers and torsion coefficients over Z of the singular homology groups of Mn, are finite and calculable. However it is not known that a “triangulation” of Mn always exists when n > 3.

Singular homology theory in the category of soft topological spaces

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Sadi Bayramov ◽  
Cigdem Gündüz (Aras) ◽  
Leonard Mdzinarishvili
Keyword(s):  
Homology Group ◽  
Homology Theory ◽  
Topological Spaces ◽  
Homology Groups ◽  
Relative Homology ◽  
Singular Homology

AbstractIn the category of soft topological spaces, a singular homology group is defined and the homotopic invariance of this group is proved [Internat. J. Engrg. Innovative Tech. (IJEIT) 3 (2013), no. 2, 292–299]. The first aim of this study is to define relative homology groups in the category of pairs of soft topological spaces. For these groups it is proved that the axioms of dimensional and exactness homological sequences hold true. The axiom of excision for singular homology groups is also proved.

scholarly journals Computing the Rabinowitz Floer homology of tentacular hyperboloids

2021 ◽  
Vol 17 (0) ◽  
pp. 353
Author(s):  
Alexander Fauck ◽  
Will J. Merry ◽  
Jagna Wiśniewska
Keyword(s):  
Floer Homology ◽  
Homology Groups ◽  
Weinstein Conjecture ◽  
Singular Homology ◽  
Rabinowitz Floer Homology ◽  
Compact Case ◽  
Chain Map ◽  
A Chain

<p style='text-indent:20px;'>We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids <inline-formula><tex-math id="M1">\begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}</tex-math></inline-formula>. Using an embedding of a compact sphere <inline-formula><tex-math id="M2">\begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document}</tex-math></inline-formula> into the hypersurface <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, we construct a chain map from the Floer complex of <inline-formula><tex-math id="M4">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> to the Floer complex of <inline-formula><tex-math id="M5">\begin{document}$ \Sigma_0 $\end{document}</tex-math></inline-formula>. In contrast to the compact case, the Rabinowitz Floer homology groups of <inline-formula><tex-math id="M6">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.</p>

scholarly journals Singular homology algorithm for MA-spaces

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2139-2147
Author(s):  
Demir Unver
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Digital Geometry ◽  
Topological Approach ◽  
Homology Groups ◽  
And Topology ◽  
Singular Homology ◽  
Digital Spaces

The work on digitizing subspaces of the 2-D Euclidean space with a certain digital approach is an important discipline in both digital geometry and topology. The present work considers Marcus-Wyse topological approach which was established for studying 2-D digital spaces, ?2. We introduce the digital singular homology groups of MA-spaces (M-topological space with an M-adjacency), and we compute singular homology groups of some certain MA-spaces, we give a formula for singular homology groups of 2-D simple closed MA-curves, and an algorithm for determining homology groups of an arbitrary MA-space.

scholarly journals Profinite rigidity for twisted Alexander polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Rigidity Theorem ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Twisted Homology ◽  
Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

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