scholarly journals Homology and K-theory of dynamical systems I. Torsion-free ample groupoids

2021 ◽  
pp. 1-31
Author(s):  
VALERIO PROIETTI ◽  
MAKOTO YAMASHITA
Keyword(s):  
Dynamical Systems ◽  
Spectral Sequence ◽  
Strong Form ◽  
Topological Dynamics ◽  
Triangulated Category ◽  
Integer Coefficients ◽  
K Theory ◽  
Torsion Free

Abstract Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C $^*$ -algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.

On the Kunneth formula spectral sequence in equivariant K- theory

1972 ◽  
Vol 72 (2) ◽  
pp. 167-177 ◽  
Author(s):  
V. P. Snaith
Keyword(s):  
Vector Bundle ◽  
Spectral Sequence ◽  
Lie Group ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Kunneth Formula ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

Let G be a compact, connected Lie group such that π2(G) is torsion free. Throughout this paper a vector bundle (representation) will mean a complex vector bundle (representation) and KG will denote the equivariant K-theory functor associabed with the group, G.

scholarly journals Category Theory for Autonomous and Networked Dynamical Systems

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 302 ◽  
Author(s):  
Jean-Charles Delvenne
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Ergodic Theory ◽  
Category Theory ◽  
Open Systems ◽  
Topological Dynamics ◽  
Control Problems ◽  
Natural Conditions ◽  
Discussion Paper ◽  
Open Systems Theory

In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.

scholarly journals On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

scholarly journals Pro unitality and pro excision in algebraic K-theory and cyclic homology

2018 ◽  
Vol 2018 (736) ◽  
pp. 95-139 ◽  
Author(s):  
Matthew Morrow
Keyword(s):  
Short Proof ◽  
Hochschild Homology ◽  
Cyclic Homology ◽  
Strong Form ◽  
Noetherian Rings ◽  
General Base ◽  
Continuity Properties ◽  
Suitable Sense ◽  
K Theory

AbstractThe purpose of this paper is to study pro excision in algebraicK-theory and cyclic homology, after Suslin–Wodzicki, Cuntz–Quillen, Cortiñas, and Geisser–Hesselholt, as well as continuity properties of André–Quillen and Hochschild homology. A key tool is first to establish the equivalence of various pro Tor vanishing conditions which appear in the literature.This allows us to prove that all ideals of commutative, Noetherian rings are pro unital in a suitable sense. We show moreover that such pro unital ideals satisfy pro excision in derived Hochschild and cyclic homology. It follows hence, and from the Suslin–Wodzicki criterion, that ideals of commutative, Noetherian rings satisfy pro excision in derived Hochschild and cyclic homology, and in algebraicK-theory.In addition, our techniques yield a strong form of the pro Hochschild–Kostant–Rosenberg theorem; an extension to general base rings of the Cuntz–Quillen excision theorem in periodic cyclic homology; a generalisation of the Feĭgin–Tsygan theorem; a short proof of pro excision in topological Hochschild and cyclic homology; and new Artin–Rees and continuity statements in André–Quillen and Hochschild homology.

The MOD 2 K-Homology of Ω3S3X

1988 ◽  
Vol 40 (1) ◽  
pp. 142-196 ◽  
Author(s):  
J. G. Mayorquin
Keyword(s):  
Spectral Sequence ◽  
Cyclic Group ◽  
Image Position ◽  
Torsion Free ◽  
Infinite Cycles

In order to compute the group K*(Ω3S3X; Z/2) when X is a finite, torsion free CW-complex we apply the techniques developed by Snaith in [38], [39], [40], [41] which were used in [42] to determine the Atiyah-Hirzebruch spectral sequence ( [11], [1, Part III])for X as above. Roughly speaking the method consists in defining certain classes in K*(Ω3S3X; Z/2) via the π-equivariant mod 2 K-homology of S2 × Y2,([35]), π the cyclic group of order 2 (acting antipodally on S2, by permuting factors in Y2, and diagonally on S2 × Y2), Y a finite subcomplex of Ω3S3X, and then showing that the classes so produced map under the edge homomorphism to cycles (in the E1-term of the Atiyah-Hirzebruch spectral sequence forwhich determine certain homology classes of H*(Ω3S3X; Z/2), thus exhibiting these as infinite cycles of the spectral sequence

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

scholarly journals C*-Algebras over Topological Spaces: Filtrated K-Theory

2012 ◽  
Vol 64 (2) ◽  
pp. 368-408 ◽  
Author(s):  
Ralf Meyer ◽  
Ryszard Nest
Keyword(s):  
Exact Sequence ◽  
Topological Space ◽  
Spectral Sequence ◽  
Topological Spaces ◽  
C Algebra ◽  
C Algebras ◽  
Finite Spaces ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Kasparov Theory

AbstractWe define the filtrated K-theory of a C*-algebra over a finite topological spaceXand explain how to construct a spectral sequence that computes the bivariant Kasparov theory overXin terms of filtrated K-theory.For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C*-algebras over a spaceXwith four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this spaceX, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

scholarly journals G2-metrics arising from non-integrable special Lagrangian fibrations

2019 ◽  
Vol 6 (1) ◽  
pp. 348-365 ◽  
Author(s):  
Ryohei Chihara
Keyword(s):  
Dynamical Systems ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Special Lagrangian ◽  
3 Dimensional ◽  
Dimensional Group ◽  
Constrained Dynamical Systems ◽  
Definite Symmetric Matrix ◽  
Type 3 ◽  
Torsion Free

AbstractWe study special Lagrangian fibrations of SU(3)-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group G, we decompose such SU(3)-structures into triples of solder 1-forms, connection 1-forms and equivariant 3 × 3 positive-definite symmetric matrix-valued functions on principal G-bundles over 3-manifolds. As applications, we describe regular parts of G2-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of G = T3 and SO(3).

scholarly journals Finitely generated nilpotent group C*-algebras have finite nuclear dimension

2018 ◽  
Vol 2018 (738) ◽  
pp. 281-298 ◽  
Author(s):  
Caleb Eckhardt ◽  
Paul McKenney
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Finitely Generated ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Torsion Free

Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

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