scholarly journals Continuous cohomology of topological quandles

2019 ◽  
Vol 28 (06) ◽  
pp. 1950036 ◽  
Author(s):  
Mohamed Elhamdadi ◽  
Masahico Saito ◽  
Emanuele Zappala
Keyword(s):  
Inverse Limit ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Algebraic Theories ◽  
Continuous Cohomology

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the inverse limit is applied to quandles.

scholarly journals A new description of equivariant cohomology for totally disconnected groups

2008 ◽  
Vol 1 (3) ◽  
pp. 431-472 ◽  
Author(s):  
Christian Voigt
Keyword(s):  
Equivariant Cohomology ◽  
Simplicial Complexes ◽  
Cyclic Homology ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Natural Class ◽  
New Approach ◽  
Totally Disconnected

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

scholarly journals Note on the Cohomology Groups of Associative Algebras

1953 ◽  
Vol 6 ◽  
pp. 85-92 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Detailed Analysis ◽  
Structural Characterization ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Associative Algebras ◽  
3 Dimensional

The cohomology theory of associative algebras has been developed by G. liochschild [1], [2], [3], and the 1-, 2-, and 3-dimensional cohomology groups have been interpreted with reference to classical notions of structure in his papers. Recently M. Ikeda has obtained, by a detailed analysis of Hochschild’s modules, an interesting structural characterization of the class of algebras whose 2-dimensional cohomology groups are all zero [5].

scholarly journals Foundations of topological racks and quandles

2016 ◽  
Vol 25 (03) ◽  
pp. 1640002 ◽  
Author(s):  
Mohamed Elhamdadi ◽  
El-Kaïoum M. Moutuou
Keyword(s):  
Automorphism Group ◽  
Cohomology Theory ◽  
Central Extensions ◽  
Continuous Cohomology ◽  
Inner Automorphism

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step toward a general continuous cohomology theory for topological racks and quandles.

scholarly journals Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields

1954 ◽  
Vol 2 (2) ◽  
pp. 66-76 ◽  
Author(s):  
Iain T. Adamson
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Cohomology Group ◽  
Cohomology Theory ◽  
Normal Subgroups ◽  
Cohomology Groups ◽  
Coset Space ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that ifis an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequencemay be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequenceis exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

The spectrum (P ⋀ bo) −∞

1984 ◽  
Vol 96 (1) ◽  
pp. 85-93 ◽  
Author(s):  
Donald M. Davis ◽  
Mark Mahowald
Keyword(s):  
Inverse Limit ◽  
Projective Spaces ◽  
Inverse System ◽  
Cohomology Groups ◽  
Image Position ◽  
Adic Completion

There are spectra P−k constructed from stunted real projective spaces as in [1] such that H*(P−k) is the span in ℤ/2[x, x−1] of those xi with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse systemwhich is similar to those of Lin ([15], p. 451). It is a corollary of Lin's work that there is an equivalence of spectrawhere holim is the homotopy inverse limit ([3], ch. 5) and Ŝ–1 the 2-adic completion of a sphere spectrum. One may denote by this holim (P–κ), although one must constantly keep in mind that , but rather

scholarly journals Comparison of the discrete and continuous cohomology groups of a pro-$p$ group

2008 ◽  
Vol 19 (6) ◽  
pp. 961-973 ◽  
Author(s):  
G. A. Fernández-Alcober ◽  
I. V. Kazachkov ◽  
V. N. Remeslennikov ◽  
P. Symonds
Keyword(s):  
Cohomology Groups ◽  
Continuous Cohomology

scholarly journals Algebras With Vanishing n-Cohomology Groups

1954 ◽  
Vol 7 ◽  
pp. 115-131 ◽  
Author(s):  
Masatoshi Ikeda ◽  
Hiroshi Nagao ◽  
Tadashi Nakayama
Keyword(s):  
Associative Algebra ◽  
Finite Rank ◽  
Unit Element ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Associative Algebras ◽  
Semisimple Algebras

Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.

Relative Cohomology

1957 ◽  
Vol 9 ◽  
pp. 19-34 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Cohomology Theory ◽  
Cohomology Groups ◽  
Dual Pairs ◽  
Relative Cohomology

It is our purpose in this paper to present certain aspects of a cohomology theory of a ring R relative to a subring S, basing the theory on the notions of induced and produced pairs of our earlier paper (2), but making the paper self-contained except for references to a few specific results of (2). The cohomology groups introduced occur in dual pairs. Generic cocycles are defined, and the groups are related to the protractions and retractions of R-modules.

scholarly journals A construction of complex analytic elliptic cohomology from double free loop spaces

2021 ◽  
Vol 157 (8) ◽  
pp. 1853-1897
Author(s):  
Matthew Spong
Keyword(s):  
Elliptic Curves ◽  
Inverse Limit ◽  
Loop Space ◽  
Equivariant Cohomology ◽  
Cohomology Theory ◽  
Free Loop Space ◽  
Loop Spaces ◽  
Elliptic Cohomology ◽  
Finite Dimensional ◽  
Moduli Stack

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$ , the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$ .

scholarly journals Higher arity self-distributive operations in Cascades and their cohomology

2020 ◽  
pp. 2150116
Author(s):  
Mohamed Elhamdadi ◽  
Masahico Saito ◽  
Emanuele Zappala
Keyword(s):  
Lie Algebras ◽  
Hopf Algebras ◽  
Geometric Interpretation ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Monoidal Categories ◽  
Link Invariants ◽  
Binary Case ◽  
Symmetric Monoidal Categories ◽  
Framed Link

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing [Formula: see text]-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.

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