group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
The construction of algebraic K-theory $K(R)$, originally defined for rings $R$, generalizes to $A_\infty$-ring spectra. When $R$ happens to be a connective $E_\infty$-ring spectrum, then also the representing spectrum $K(R)$ of its algebraic K-theory is a connective $E_\infty$-ring spectrum (Schwänzl & Vogt 1994, Thm. 1, EKMM 1997, Thm. 6.1) so that this construction may then be iterated (Rognes 2014) to yield iterated algebraic K-theories $K(K(R))$, $K(K(K(R)))$, etc.
The red-shift conjecture says that this iteration plays a special role in chromatic homotopy theory.
The construction of iterated algebraic K-theory has received particular attention for the case that $R =$ ku is the connective ring spectrum representing complex topological K-theory.
Here the first iterated stage $K(ku)$ is related to BDR 2-vector bundles essentially like ku is related to ordinary complex vector bundles.
The tower $K^{2r}(ku)$ of higher iterated algebraic K-theories of topological K-theory has been shown to accommodate a generalization of the Fourier-Mukai-type transform on twisted K-theory that is given by topological T-duality, generalizing it to spherical T-duality (Lind-Sati-Westerland 16).
See at red-shift conjecture.
On the algebraic K-theory of (connective) ring spectra:
Roland Schwänzl, Rainer Vogt, Basic Constructions in the K-Theory of Homotopy Ring Spaces, Transactions of the American Mathematical Society, Vol. 341, No. 2 (Feb., 1994), pp. 549-584 (jstor:2154572, doi:10.2307/2154572)
Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter VI of Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)
Andrew Blumberg, David Gepner, Gonçalo Tabuada, Section 9.5 of: A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013) 733-838 (arXiv:1001.2282, doi:10.2140/gt.2013.17.733)
Jacob Lurie, Algebraic K-Theory of Ring Spectra, Lecture 19 of Algebraic K-Theory and Manifold Topology, 2014 (pdf)
The algebraic K-theory specifically of suspension spectra of loop spaces (Waldhausen’s A-theory) is originally due to
On the iteration of the construction and the red-shift conjecture:
On the algebraic K-theory $K(R)$ of a ring spectrum $R$ as the Grothendieck group of (∞,1)-module bundles over $R$:
On the first algebraic K-theory $K(ku)$ of connective topological K-theory:
Christian Ausoni, On the algebraic K-theory of the complex K-theory spectrum, Inventiones mathematicae volume 180, pages 611–668 (2010) (arXiv:0902.2334, doi:10.1007/s00222-010-0239-x)
Christian Ausoni, John Rognes, Algebraic K-theory of topological K-theory, Acta Math. Volume 188, Number 1 (2002), 1-39 (euclid:acta/1485891473)
Christian Ausoni, John Rognes, Rational algebraic K-theory of topological K-theory, Geom. Topol. 16 (2012) 2037-2065 (arXiv:0708.2160, doi:10.2140/gt.2012.16.2037)
Interpretation of $K(ku)$ as the K-theory of BDR 2-vector bundles:
Nils Baas, Bjørn Ian Dundas, John Rognes, Two-vector bundles and forms of elliptic cohomology, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004 (arXiv:math/0306027, doi:10.1017/CBO9780511526398.005)
Nils Baas, Bjørn Ian Dundas, Birgit Richter, John Rognes, Stable bundles over rig categories, Journal of Topology, Volume 4, Issue 3, September 2011, Pages 623–640 (arXiv:0909.1742, doi:10.1112/jtopol/jtr016)
which can also be understood as a special case of K-theory for 2-categories:
On the algebraic K-theory of algebraic K-theory of finite fields $K(K(\mathbb{F}))$:
Discussion of higher and of twisted iterated K-theory on $ku$, and realization of the spherical T-duality on twisted $K^{2r}(ku)$:
Last revised on March 20, 2024 at 08:18:08. See the history of this page for a list of all contributions to it.