algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The homological version of what is called the group completion theorem (McDuff & Segal 1976) relates the Pontrjagin ring of a topological monoid $A$ to that of its group completion $\Omega B A$.
The original articles are
Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici 47 1 (1972) [doi:10.1007/BF02566785]
Dusa McDuff, Graeme Segal, Homology fibrations and the “group-completion” theorem, Inventiones mathematicae 31 3 (1976) 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:
Alternative formulation for the case of commutative topological monoids:
Oscar Randal-Williams, Group-completion, local coefficient systems and perfection, Q. J. Math. 64 3 (2013) 795–803 [pdf, doi:10.1093/qmath/hat024]
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:
A proof based on Nikolaus’s proof was written up in
Last revised on July 24, 2024 at 20:15:33. See the history of this page for a list of all contributions to it.