The homotopy groups of a homotopy group completion
type: Item
instance of: scholarly article
title: The homotopy groups of a homotopy group completion
zbMATH Open document ID: 1461.19002
published in: Israel Journal of Mathematics
publication date: 2019-12-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1807.02613
review text: Let \(M\) be a topological monoid, \(BM\) the classifying space of \(M\), \(\Omega BM\) the (based) loop space of \(BM\), and \([S^k,M]\) the set of (unbased) homotopy classes of (continuous) maps between the \(k\)-sphere \(S^k\) and \(M\) (which is naturally a monoid). Say that \(M\) satisfies the \textit{strong homotopy commutativity hypothesis} (SHCH) if for every \(x\in M\) there exists a homotopy between the maps \((m,n)\mapsto mn\) and \((m,n)\mapsto nm\) that is constant on \((x,x)\). In this well-written and detailed paper, it is proved that if \(M\) satisfies SHCH, then for any \(k>0\), there is a natural isomorphism of groups: \[[S^k,M]/\pi_0(M)\cong \pi_k(\Omega BM).\] The isomorphism is described explicitly in the proof. A priori, it is not immediately clear to this reviewer that the quotient monoid \([S^k,M]/\pi_0(M)\) is a group to begin with; part of the proof of this result establishes the existence of inverses in \([S^k,M]/\pi_0(M)\). The result seems both powerful and interesting. Indeed, the author applies this theorem to numerous settings where topological monoids naturally arise and have been the subject of active study: the Lawson homology of a complex projective variety, the deformation representation ring of discrete groups, the deformation \(K\)-theory of discrete groups, and lastly to the lifting problem of (spherical) families of characters to families of representations. With regard to this last application, let \(\Gamma\) be a finitely generated discrete group and let \(G\) be a connected real reductive Lie group. The \(G\)-character variety of \(\Gamma\) is the conjugation quotient space \(\mathrm{X}(\Gamma, G):=\mathrm{Hom}(\Gamma, G)^*/G\) where \(\mathrm{Hom}(\Gamma, G)^*\subset \mathrm{Hom}(\Gamma, G)\) is the subspace of group homomorphisms whose \(G\)-conjugation orbits are closed. A spherical family is a continuous map \(\chi:S^k\to \mathrm{X}(\Gamma, G)\). Assume now that \(G_n\) is either \(\mathrm{GL}(n,\mathbb{C})\) or \(\mat…
reviewed by: Sean Lawton
Mathematics Subject Classification ID: 19A99
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 14M35
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55N25
zbMATH DE Number: 7143807
zbMATH Keywords: \(K\)-theory, topological monoid, homotopy groups, character variety
author: Daniel Ramras
MaRDI profile type: Publication
OpenAlex ID: W2969651692
arXiv ID: 1807.02613
cites work: The topology of nilpotent representations in reductive groups and their maximal compact subgroups.
cites work: Q4780265
cites work: The local structure of algebraic K-theory
cites work: Semisimplicial spaces
cites work: Group-like structures in general categories. I. Multiplications and comultiplications
cites work: Q4716283
cites work: The topology of moduli spaces of free group representations
cites work: Topology of character varieties of abelian groups
cites work: Homotopy groups of free group character varieties
cites work: Q3970368
cites work: Q4819371
cites work: Q4097031
cites work: The topological structure of the space of algebraic varieties
cites work: The product formula in unitary deformation \(K\)-theory
cites work: The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)
cites work: Q4650609
cites work: Varieties of representations of finitely generated groups
cites work: Homology fibrations and the 'group-completion' theorem
cites work: A TWISTED HOMOLOGY FIBRATION CRITERION AND THE TWISTED GROUP-COMPLETION THEOREM
cites work: The geometric realization of a semi-simplicial complex
cites work: Q4650608
cites work: A finite-dimensional approach to the strong Novikov conjecture
cites work: Excision for deformation \(K\)-theory of free products
cites work: The stable moduli space of flat connections over a surface
cites work: Periodicity in the stable representation theory of crystallographic groups
cites work: Yang-Mills theory over surfaces and the Atiyah-Segal theorem
cites work: `Group-completion', local coefficient systems and perfection
cites work: Categories and cohomology theories
cites work: Geometric realization and its variants
DOI: 10.1007/S11856-019-1914-2
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description: scientific article