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.

Three-taxon statement analysis (3TA) and standard cladistic analysis (SCA) were evaluated relative to propositions of taxic homology. There are definite distinctions between complement relation homologs and paired homologs. The complement relation is discussed, relative to rooting, parsimony, and taxic propositions of homology. The complement relation, as implemented in SCA, makes sense only be...

In an earlier paper we have shown that the pair-of-pants product on the Floer homology of the cotangent bundle of an oriented compact manifold Q corresponds to the Chas-Sullivan loop product on the singular homology of the free loop space of Q. We now give chain level constructions of further product structures in Floer homology, corresponding to the cup product on the homology of any path spac...

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements’ size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(βn log n...

For an extension of associative algebras B ⊂ A over a field and A-bimodule X, we obtain Jacobi–Zariski long nearly exact sequence relating the Hochschild homologies B, relative homology, all them with coefficients in X. This is twice three. There spectral which converges to gap exactness.

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in t...

If H is a finitely generated subgroup of the finitely generated group G, then H is undistorted in G for their word metrics iff the induced homomorphism on the zero-dimensional l1-homology group is injective. The Mayer-Vietoris exact sequence for l1-homology is established and applications are given to hyperbolic groups and relative hyperbolicity. The notion of relative distortion of two maps is...

We report an ongoing attempt to establish in algebraic geometry certain analogues of topological ideas, The main goal is to associate to a scheme X over a commutative ring k its “relative motivic homology” which is again an algebro geometric object over the base k. This is motivated by Number Theory, so the Poincare duality for this relative motivic homology should be an algebro geometric incar...