Poincaré duality for $$L^p$$ cohomology on subanalytic singular spaces

We investigate the problem of Poincaré duality for $$L^p$$ L p differential forms on bounded subanalytic submanifolds of $$\mathbb {R}^n$$ R n (not necessarily compact). We show that, when p is sufficiently close to 1 then the $$L^p$$ L p cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that $$L^p$$ L p cohomology is dual to intersection homology. As a consequence, we can deduce that the $$L^p$$ L p cohomology is Poincaré dual to $$L^q$$ L q cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large.

Computing the Rabinowitz Floer homology of tentacular hyperboloids

<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 algorithm for MA-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.

EXOTIC FINITE FUNCTORIAL SEMI-NORMS ON SINGULAR HOMOLOGY

Functorial semi-norms on singular homology give refined ‘size’ information on singular homology classes. A fundamental example is the ℓ1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they arenotcarried by the ℓ1-semi-norm.