We develop a method for measuring homology classes. This involves two 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(βn3 log n)...

We introduce here a new F2 homology computation algorithm based on a generalization of the spanning tree technique on a finite 3-dimensional cell complex K embedded in R. We demonstrate that the complexity of this algorithm is linear in the number of cells. In fact, this process computes an algebraic map φ over K, called homology gradient vector field (HGVF), from which it is possible to infer ...

Braid Floer homology is an invariant of proper relative braid classes [12]. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid classes is non-trivial, then additional closed integral curves of the Hamiltonian equations are forced via a Morse type theory. In this article we show t...

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 ...

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...

where s is a relative Spinc structure for B̃ and [F ] is the homology class of a pre-image of a spanning disc for B. The author began studying this situation in [14] where a connection to Kohvanov homology is described. In this paper, we wish to use the same approach to study the special case where L is a braid in the complement of B, and derive the complete knot Floer homology for a myriad of f...

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 ...

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be proc...

We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincaré duality hypothesis, such as CalabiYau algebras, derived Poincaré duality algebras and closed Frobenius algebras. This includes a BV-algebra structure on HH(A,A) or HH(A,A), which in the latter case is an extension of the natural Gerstenhaber structure on HH(A,A...

This is the second part of a series papers which bridges Chang–Weinberger–Yu relative higher index and geometry almost flat Hermitian vector bundles on manifolds with boundary. In this paper, we apply description given in I to establish version Hanke–Schick theorem, relates pairing K-homology cycles bundles. We also deal quantitative dual problem theorem.