To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from Khovanov homology boundary link. Previous work showed that when link is empty, this determined by genus surface. We show relative invariant: can obstruct sliceness knots; detects pair slices for $9_{46}$; not hindered detecting connected sums with knott...

Let p be a prime number, and let A ring in which is nilpotent . In this paper, we consider the maps K q + 1 ( [ x ] / m ) , → n induced by homomorphism ↦ We evaluate these maps, up to extension, for general terms of topological Hochschild homology, regular F -algebras groups de Rham-Witt forms. After evaluation, give calculation relative -group O certain perfectoid fields

This is a further investigation of our approach to group actions in homological algebra the settings homology $$\Gamma $$ -simplicial groups, particularly -equivariant and cohomology -groups. new direction called -homological algebra. The abstract kernel non-abelian extensions its relationship with obstruction theory, second are extended case -extensions We compute rational (co)homology groups ...

In this paper, we define the homological Morse numbers of a filtered cell complex in terms relative homology nested filtration pieces, and derive inequalities relating these to Betti tables multi-parameter persistence modules considered filtration. Using Mayer-Vietoris spectral sequence first obtain strong weak involving above quantities, then improve achieving sharp lower bound for numbers. Fu...

In [5], we computed the cyclic homology of finite extensions of Dedekind domains. The crucial result was a formula for the cyclic homology, relative to a ground ring R, of A = R[x]/(P (x)). This paper presents two extensions of the methods and results of this paper. Recall that Hochschild homology, though generally defined in terms of a bar complex, is almost never, in practice, computed from i...

We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions orbi-CW-pairs by vector bundles and morphisms classes relative maps. The representable orbispectra admits a contravariant involution extending Spanier--Whitehead duality. This duality relates homotopical cobordism theories (cohomology on orbispectra) represented global Thom sp...

This thesis deals with the homology and cohomology groups of simplicial complexes, and especially with the two duality theorems which willdemonstrate links between the two notions. For this we will need some results, and we will therefore prove the long exact sequence for the reduced relative homology group; and to link the homology and cohomolgy to our geometric notion of a simplicial complex ...

The Lawrence-Krammer representation of the braid groups recently came to prominence when it was shown to be faithful by myself and Krammer. It is an action of the braid group on a certain homology module H2(C̃) over the ring of Laurent polynomials in q and t . In this paper we describe some surfaces in C̃ representing elements of homology. We use these to give a new proof that H2(C̃) is a free mod...

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several directions: the relative weight filtration of Dennis Johnson, the finite-type ...