in this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. we present sufficient conditions for the existence of long exact extended d- homology sequence. also we give some illustrative examples.

Abstract We construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to familiar package of homology cohomology. In order do this conceptually, we sketch a version calculus , sense [12], adapted context algebras. obtain standard theorems (co)homology chiral context. This is part larger project related promoting noncomm...

The ?ech and Rips constructions of persistent homology are stable with respect to perturbations the input data. However, neither is robust outliers, both can be insensitive topological structure high-density regions A natural solution consider 2-parameter persistence. This paper studies stability homology: we show that several related density-sensitive bifiltrations from data satisfy properties...

In this article, we study the magnitude homology of geodesic metric spaces curvature $\leq \kappa$, especially ${\rm CAT}(\kappa)$ spaces. We will show that $MH^{l}_{n}(X)$ such a meric space $X$ vanishes for small $l$ and all $n > 0$. Conseqently, can compute total $\mathbb{Z}$-degree shperes $\mathbb{S}^{n}$, Euclid $\mathbb{E}^{n}$, hyperbolic $\mathbb{H}^{n}$, real projective $\mathbb{RP}^{...