A classifying space for commutativity in Lie groups

Abstract

Abstract. In this article we consider a space B com G assembled from commuting elements in a Lie group G first defined in [3]. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that Z × B com U is a loop space and define a notion of commutative K-theory for bundles over a finite complex X which is isomorphic to [X, Z × B com U ]. We compute the rational cohomology of B com G for G equal to any of the classical groups SU (r), U (q) and Sp(k), and exhibit the rational cohomologies of B com U , B com SU and B com Sp as explicit polynomial rings.

Commutative cocycles and stable bundles over surfaces

“…The effect of post-composing with φ 2 can be seen at the level of simplicial spaces. (2). Writing this factorization as j • h k , it suffices to show that [h k ] = [g k ] in π 2 (BSO(2)).…”

“…To prove part (2), first notice that part (1) and Proposition 4.1 for n = 3 imply that π 2 (B com O(2)) → π 2 (B com O(3)) is surjective. The isomorphism (7) and surjectivity at π 2 of the inclusions B com O(n) → B com O(n+1) imply that the groups π 2 (B com O(n)) are all quotients of Z/2 ⊕ Z/2.…”

“…Let X be a finite CW complex. In Adem-Gómez [2], commuting orthogonal matrices were used to construct a variant of real topological K-theory, KO com (X), whose classes are represented by vector bundles over X equipped with a commutative trivialization. A trivialization of a bundle E → X is called commutative if all pairs of transition functions commute (as elements of the structure group) wherever they are simultaneously defined (see Definition 2.1).…”

“…[2],Adem, Gómez). Let X be a finite CW-complex and E → X a principal G-bundle classified by f : X → BG.…”

Commuting matrices and Atiyah's real K‐theory

“…The horizontal axis is a copy of the ring π * (ko) ∼ = Z[η, x, y]/(2η, ηx, η 3 , x 2 − 4y), where η = av generates π 1 (ko), x = wv 2 generates π 4 (ko) and y = Uv 4 generates π 8 (ko). Passing to C 2 -fixed points, the splitting (5) shows that there is an isomorphism…”