Singular homology groups of one-dimensional Peano continua

2016 ◽  
Vol 232 (2) ◽  
pp. 99-115 ◽  
Author(s):  
K. Eda
Keyword(s):  
One Dimensional ◽  
Homology Groups ◽  
Peano Continua ◽  
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>

Singular Homology of One-Dimensional Spaces

1959 ◽  
Vol 69 (2) ◽  
pp. 309 ◽  
Author(s):  
M. L. Curtis ◽  
M. K. Fort
Keyword(s):  
One Dimensional ◽  
Singular Homology
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