nLab group completion theorem

Idea

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

References

The original articles are

• Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici (1972) 47: 1 (doi:10.1007/BF02566785)

• Dusa McDuff, Graeme Segal, Homology fibrations and the “group-completion” theorem, Inventiones mathematicae, October 1976, Volume 31, Issue 3, pp 279–284 (doi:10.1007/BF01403148)

• Daniel Quillen, On the group completion of a simplicial monoid, Appendix Q in: Eric Friedlander, Barry Mazur, Filtrations on the homology of algebraic varieties, Memoir of the A.M.S., Vol. 110, no. 529 (1994) (doi:10.1090/memo/0529, pdf)

Alternative proof using a model category of bisimplicial sets:

• Ieke Moerdijk, Bisimplicial Sets and the Group-Completion Theorem In: Rick Jardine, Victor Snaith (eds.) Algebraic K-Theory: Connections with Geometry and Topology, NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 279. Springer, Dordrecht (1989) (doi:10.1007/978-94-009-2399-7_10)

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)

A proof based on Nikolaus’s proof was written up in

• Oscar Bendix Harr, Group completion is a completion, PDF.