group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
Orbifold K-theory is K-theory (typically: topological K-theory) of orbifolds, typically meant to reduce to equivariant K-theory on global quotient orbifolds.
In general it is subtle to decide whether a given orbifold cohomology-theory $E^\bullet(\mathcal{X})$ is equivalently the $G$-equivariant cohomology of a topological G-space $X$ for any realization of $\mathcal{X}$ as a global quotient orbifold of $X$ by $G$ (as highlighted inPronk-Scull 07). But for topological equivariant K-theory this is the case, by Pronk-Scull 07, Prop. 4.1.
Therefore it makes sense to define the K-theory of an orbifold $\mathcal{X}$ which is equivalent to a global quotient orbifold
to be the $G$-equivariant K-theory of $X$:
This is the approach taken in AdemRuan 01, Def. 3.4
The definition originates, via bundle gerbes and bundle gerbe modules on Lie groupoids, in:
The definition of the K-theory of global quotient orbifolds as the twisted equivariant K-theory of the universal covering space appears in
Review in
The general proof that this is well-defined (independent of the realization of the orbifold as a global quotient) is due to
A more geometric model of orbifold K-theory in terms of bundles of Fredholm operators over Lie groupoids/differentiable stacks:
Review in:
Daniel Freed, Lecture 1 of: Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (pdf)
Joost Nuiten, Section 3.2.2 of: Cohomological quantization of local prequantum boundary field theory MSc thesis, Utrecht, August 2013 (pdf)
Valentin Zakharevich, Sections 2.2, 2.3 of: K-Theoretic Computation of the Verlinde Ring, thesis 2018 (hdl:2152/67663, pdf, pdf)
The claim that these two definitions are equivalent, in that this groupoid K-theory reduces to equivariant K-theory on global quotient orbifolds, is Freed-Hopkins-Teleman 07, Prop. 3.5.
Another definition of K-theory of orbifolds (“full orbifold K-theory”) is due to
proven there to coincide with Adem-Ruan 01 on global quotients.
The suggestion (Schwede 17, Intro, Schwede 18, p. ix-x) that orbifolds should be regarded as orbispaces in global equivariant homotopy theory and then their orbifold cohomology be given by equivariant cohomology with coefficients in global equivariant spectra is worked out for (Bredon cohomology and) orbifold K-theory in:
Example 5.31 there shows that on global quotient orbifolds this is again equivalent to the previous definitions.
Alejandro Adem, Yongbin Ruan, Bin Zhang, A Stringy Product on Twisted Orbifold K-theory, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64. (arXiv:math/0605534, Morfismos pdf)
Edward Becerra, Bernardo Uribe, Stringy product on twisted orbifold K-theory for abelian quotients, Trans. Amer. Math. Soc. 361 (2009), 5781-5803 (arXiv:0706.3229, doi:10.1090/S0002-9947-09-04760-6)
Jianxun Hu, Bai-Ling Wang, Delocalized Chern character for stringy orbifold K-theory, Trans. Amer. Math. Soc. 365 (2013), 6309-6341 (arXiv:1110.0953, doi:10.1090/S0002-9947-2013-05834-5)
Last revised on December 11, 2022 at 14:53:07. See the history of this page for a list of all contributions to it.