group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
What is called symplectic K-theory (denoted $K Sp$) or, equivalently, quaternionic K-theory (denoted $K Q$) is the topological K-theory of quaternionic vector bundles.
This is in direct analogy with how complex K-theory (KU) is the topological K-theory of complex vector bundles and KO that of real vector bundles.
In fact, quaternionic K-theory is equivalently KO-theory in degrees which are multiples of 4.
(…)
In solid state physics, under the K-theory classification of topological phases of matter, quaternionic K-theory is thought to classify crystalline topological insulator-phases of materials whose electron-dynamics respects time-reversal symmetry (this makes the classification be by $KO^\bullet$) but no parity symmetry? (this restricts the classification further to $KO^{4 n} \simeq K Q^n$). Then spin-less such phases are classified by $K Q^0 \simeq K O^0$, while spin-ful such phases are classified by $K Q^1$.
In this context, the Kane-Mele invariant is a projection $K Q^1(\mathbb{T}^d) \to \mathbb{Z}/2$ from the quaternionic K-theory of the Brillouin torus of a material, which detects (in particular, for $d = 2$) the non-triviality of the graphene-phase (which is a time-reversal-symmetric topological insulator when its spin-orbit coupling is taken into account).
Created on May 7, 2022 at 09:43:26. See the history of this page for a list of all contributions to it.