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 ring spectra. But algebraic K-theory is itself represented by a ring spectrum, so that this construction may then be iterated 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).
On the algebraic K-theory of ring spectra:
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 of specifically of suspension spectra of loop spaces (Waldhausen’s A-theory) is originally due to
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)
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 September 4, 2020 at 09:33:38. See the history of this page for a list of all contributions to it.