Commutative $d$-torsion $K$-theory and its applications

Abstract: Commutative $K$-theory is introduced by Adem-Gomez-Lind-Tillmann as a generalized cohomology theory obtained from topological $K$-theory. The construction uses classifying spaces for commutativity, first introduced by Adem-Cohen-Torres Giese. In this talk we are interested in a $d$-torsion version of this construction: Let $G$ be a topological group. The aforementioned classifying space $B(\mathbb{Z}/d,G)$ is assembled from tuples of pairwise commuting elements in $G$ whose order divides $d$. We will describe the homotopy type of this space when $G$ is the stable unitary group, following the ideas of Gritschacher-Hausmann. The corresponding generalized cohomology theory will be called the commutative $d$-torsion $K$-theory, and will be denoted by $k\mu_d$. Our motivation for studying this cohomology theory comes from applications to operator-theoretic problems that arise in quantum information theory. For this we introduce another spectrum obtained from $k\mu_d$ and show that a famous construction from the study of quantum contextuality, known as Mermin's square, corresponds to a non-trivial class in this generalized cohomology theory. This refines the topological approach to quantum contextuality developed earlier jointly with Raussendorf.

For a related talk see www.youtube.com/watch?v=XCTHaASjurg

algebraic topologycategory theorygroup theoryK-theory and homology

Audience: researchers in the topic

( paper | video )

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Cihan Okay

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Recordings of talks available at www.youtube.com/channel/UCLrmyGpqxyeVpTcA1b5HcMw/videos

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