Applications of stable classifying space theory to group cohomology

1997 ◽  
pp. 461-473
Author(s):  
Stewart Priddy
Keyword(s):  
Group Cohomology ◽  
Space Theory ◽  
Classifying Space

scholarly journals SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION

2019 ◽  
Vol 71 (1) ◽  
pp. 207-246
Author(s):  
Bertrand J Guillou ◽  
J Peter May ◽  
Mona Merling ◽  
Angélica M Osorno
Keyword(s):  
Loop Space ◽  
General Category ◽  
Space Theory ◽  
Classifying Space ◽  
General Internal ◽  
Categorical Framework ◽  
Infinite Loop ◽  
Definition Of ◽  
A Finite Group ◽  
Infinite Loop Space

Abstract We give an operadic definition of a genuine symmetric monoidal $G$-category, and we prove that its classifying space is a genuine $E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $G$-categories to genuine permutative $G$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $G$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $G$-categories as input to an equivariant infinite loop space machine that gives genuine $\Omega $-$G$-spectra as output.

scholarly journals Nilpotence in group cohomology

2012 ◽  
Vol 56 (1) ◽  
pp. 151-175
Author(s):  
Nicholas J. Kuhn
Keyword(s):  
Lie Group ◽  
Group Cohomology ◽  
Compact Lie Group ◽  
Classifying Space ◽  
Subgroup Structure

AbstractWe study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds's recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p-groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p-group P.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

scholarly journals Improving KKM Theory on General Metric Type Spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 1-12
Author(s):  
Sehie Park
Keyword(s):  
Convex Space ◽  
Metric Spaces ◽  
Space Theory ◽  
Type Space ◽  
Generalized Metric ◽  
Abstract Convex Space ◽  
Type Spaces

Abstract A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.

Modeling learning in knowledge space theory through bivariate Markov processes

2021 ◽  
Vol 103 ◽  
pp. 102549
Author(s):  
Pasquale Anselmi ◽  
Luca Stefanutti ◽  
Debora de Chiusole ◽  
Egidio Robusto
Keyword(s):  
Markov Processes ◽  
Space Theory ◽  
Knowledge Space ◽  
Knowledge Space Theory

scholarly journals Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Siran Li
Keyword(s):  
Banach Space ◽  
Discrete Geometry ◽  
Thin Shell ◽  
Euclidean Geometry ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Space Theory ◽  
Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

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