nLab
group completion theorem

Contents

Context

Algebraic topology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The homological version of what is called the group completion theorem (McDuff-Segal 76) relates the Pontrjagin ring of a topological monoid AA to that of its group completion ΩBA\Omega B A.

References

The original articles are

Alternative proof using a model category of bisimplicial sets:

Alternative formulation for the case of commutative topological monoids:

  • Oscar Randal-Williams, Group-completion, local coefficient systems and perfection, Q. J. Math. 64 (2013), no. 3, 795–803.

  • Simon Gritschacher, A remark on the group-completion theorem (arxiv:1709.02036)

Generalization to generalized homology represented by ring spectra and relation to the Quillen plus construction:

  • Thomas Nikolaus, The group completion theorem via localizations of ring spectra, 2017 (pdf)

Last revised on October 18, 2019 at 06:08:31. See the history of this page for a list of all contributions to it.