By using surgery techniques, we compute Floer homology for certain classes of integral homology 3-spheres homology cobordant to zero. We prove that Floer homology is two-periodic for all these manifolds. Based on this fact, we introduce a new integer valued invariant of integral homology 3-spheres. Our computations suggest its homology cobordism invariance.

We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd homology. use a mapping cone construction instead of tensor product, which allows us deal efficiently with the more complicated sign assignments theory. The has been implemented computer program. also determine 3-strand torus links.

We consider the following situation: a field k, a commutative k-algebra R and a left R-module M. Since R is commutative, M may also be considered as an R-R bimodule with the same operation on each side (such modules are often termed symmetric). With these assumptions we have the Harrison (co-) homology groups Harr,(R, M) (Harr*(R, M)), the Hochschild (co-) homology groups Hoch, (R, M) (Hoch*(R,...

We use the Dolbeault cohomology to investigate Koszul–Brylinski homology on holomorphic Poisson manifolds. obtain Leray–Hirsch theorem for Hochschild and Mayer–Vietoris sequence, Künneth homology. In particular, we show some relations of homologies around a blow-up transformation general case (not necessarily compact) by our previous works cohomology.

In this paper, the authors study further properties and applications of weighted homology persistent homology. The Mayer-Vietoris sequence generalized Bockstein spectral for are introduced. For applications, show an algorithm to construct a filtration simplicial complexes from network. They also prove theorem calculate mod p2 provided with some information on p

We prove that the skein categories of Walker–Johnson-Freyd satisfy excision. This allows us to conclude are k-linear factorisation homology and taking free cocompletion recovers locally finitely presentable homology. An application this is algebra a punctured surface related any quantum group with generic parameter gives quantisation associated character variety.

We construct a minimal projective bimodule resolution for every finite dimensional quantum complete intersection of codimension two. Then we use this resolution to compute both the Hochschild cohomology and homology for such an algebra. In particular, we show that the cohomology vanishes in high degrees, while the homology is always nonzero.

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets—persistent homology—and a novel representation of this algebraic characterization— barcodes. We sketch an applica...

We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa’s formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.