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
Hochschild cohomology, cyclic 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 76) 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 (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:
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:
A proof based on Nikolaus’s proof was written up in
Last revised on May 28, 2021 at 17:44:42. See the history of this page for a list of all contributions to it.