A link surgery spectral sequence in monopole Floer homology

To a link L⊂S³, we associate a spectral sequence whose E² page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages Ek for k?2 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the δ-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well adapted to Khovanov homology.More generally, we construct new bigraded invariants of a framed link in a 3-manifold as the pages of a spectral sequence modeled on the surgery exact triangle. The differentials count monopoles over families of metrics parameterized by permutohedra. We utilize a connection between the topology of link surgeries and the combinatorics of graph-associahedra. This also yields simple realizations of permutohedra and associahedra, as refinements of hypercubes.     

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.     

Measure homology and singular homology are isometrically isomorphic

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the ℓ¹-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology – as already claimed by Thurston. For example, measure homology can be used to prove Gromov's proportionality principle of simplicial volume.     

On the Heegaard Floer homology of branched double-covers

Let be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S³, branched along L. When L is an alternating link, of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E² term is a suitable variant of Khovanov's homology for the link L, converging to the Heegaard Floer homology of Σ(L).     

The mod 2 Homology of the General Linear Group of a 2-adic Local Field

Let F be a finite extension of the 2-adic rational numbers. We compute the mod 2 homology of the general linear group GL(F) as a Hopf algebra over the Steenrod algebra. The answer is formulated in terms of the well-known homology algebras of the infinite unitary group U, its classifying space BU, and the classifying space BO of the infinite orthogonal group.     

非单位κ-代数的局部循环同调及切除定理

本文定义了单位过滤k-代数和非单位过滤k-代数的局部Hochschild同调和局部循环同调,给出 了它们之间的局部Connes长正合列．进一步利用循环同调来计算局部循环同调的短正合列公式,讨论 了关于过滤k-代数局部循环同调的切除定理．     

On knot Floer homology and lens space surgeries

In an earlier paper, we used the absolute grading on Heegaard Floer homology HF⁺ to give restrictions on knots in S³ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information can in turn be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p,q) which arise as integral surgeries on knots in S³ with |p|?1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.     

Homology groups of semicubical sets

We study the homology groups of semicubical sets with coefficients in the homological systems of abelian groups. The main theorem states that the groups under consideration are isomorphic to the homology groups of the category of singular cubes. This yields an isomorphism criterion for the homology groups of semicubical sets, the spectral sequence of a locally directed covering, and the spectral sequence of a morphism of semicubical sets.     

Dihedral and quaternionic homology and mapping spaces

We associate to the singular chains of the Moore loop space Y two dihedral chain complexes. The dihedral homology of these complexes is shown to be isomorphic to the O(2)-equivariant homology of the mapping spaces Y ^o(2) and . We also construct simple homotopy theoretic models of EO(2) x _O2 (Y)^G/BO(2) for G = O(2) or S¹, where denotes suspension. Quaternionic analogues of these results are also established.     

无限箭图的同调群

对任意箭图Q,我们研究路代数A=kQ的Hochschild同调群H_n(A)和同调群Tor_n^Ae(A,A),其中A^e是代数A的包络代数。在本文中,我们具体地给出了各次同调群和Hochschild同调群。     

On the Cyclic Homology of an Algebra Associated with a Cyclic Quiver and a Monic Polynomial

In this article, we compute the Hochschild homology group of A = KΓ/(f(X ^s )), where KΓ is the path algebra of the cyclic quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K, and X is the sum of all arrows in KΓ. Moreover, we compute the cyclic homology group of A in the case f(x) = (x ? a) ^m , where a ∈ K, so that we can determine the cyclic homology of A in general when K is an algebraically closed field.     

A Comparison of Leibniz and Cyclic Homologies

We relate Leibniz homology to cyclic homology by studying a map from a long exact sequence in the Leibniz theory to the Connes' periodicity (ISB) exact sequence in the cyclic theory. We then show that the Godbillon–Vey invariant, as detected by the Leibniz homology of formal vector fields, maps to the Godbillon–Vey invariant as detected by the cyclic homology of the universal enveloping algebra of these vector fields. Additionally the Leibniz theory maps surjectively to string topology where the latter is expressed as cyclic homology.     

On the cohomology of highly connected covers of finite Hopf spaces

Relying on the computation of the André-Quillen homology groups for unstable Hopf algebras, we prove that if the mod p cohomology of both the fiber and the base in an H-fibration is finitely generated as algebra over the Steenrod algebra, then so is the mod p cohomology of the total space. In particular, the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.     

(Co)Homology Theories for Oriented Algebras

We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally, we also study the fundamental groups of the bound quivers involved in the pullbacks.     

Some algebraic properties of cyclic homology groups

In [10], see also [8], a cyclic homology theory HC _* ^– was introduced. The purpose of this paper is to study algebraically the properties of this version of cyclic homology. First we study its relation to Connes cyclic cohomology theory HC ^* and to the usual cyclic homology theory HC _* studied by Loday and Quillen in [15]. We explain the precise sense in which HC _* ^– is dual to HC ^*. Next we study products and describe a general method for constructing product operations in cyclic homology and cohomology theories. Finally we examine the relation between the theory HC _* ^– and algebraic K-theory.     

G‐invariant persistent homology

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self‐homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo‐distance d_G associated with a group G ? Homeo(X), we need to adapt persistent homology and consider G‐invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G‐invariant persistent homology with respect to the natural pseudo‐distance d_G. We also show how G‐invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self‐homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

Pseudocycles and integral homology

We describe a natural isomorphism between the set of equivalence classes of pseudocycles and the integral homology groups of a smooth manifold. Our arguments generalize to settings well-suited for applications in enumerative algebraic geometry and for construction of the virtual fundamental class in the Gromov-Witten theory.

     

Cyclic homology of algebras with one generator

We compute the cyclic homology of A = k[X]/X ⁿ for an arbitrary commutative ring k, and we apply this result to compute the cyclic homology of k[X]/f, when k is a field and f is an arbitrary polynomial.This work was presented at the IX-ELAM, Santigo de Chile, July 1988, and was partially supported by CONICET.The following pepole participated in this research: Jorge A. Guccione, juan José guccione, Maria Julia Redondo, Andrea Solotar, and Orlando E. Villamayor.