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.

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave certain functorial ways with respect to decomposable Lagrangian cobordisms symplectization standard contact structure on R3. Our results give new, computable, effective obstructions existence such cobordisms.

We compute the first two symplectic quadratic K-theory groups of integers, or equivalently, stable homology group integral matrices preserving standard refinement. The main novelty in our calculation lies its method, which is based on high-dimensional manifold theory.

The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...

We investigate the relative cohomology and relative homology theories of $F$-Gorenstein modules, consider the relations between classical and $F$-Gorenstein (co)homology theories.

We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of t...

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S 1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the delicate transversal-ity problems in the standard approach. We also define a ...

Using Furuta’s idea of finite dimensional approximation in Seiberg–Witten theory, we refine Seiberg–Witten Floer homology to obtain an invariant of homology 3–spheres which lives in the S1–equivariant graded suspension category. In particular, this gives a construction of Seiberg–Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a re...