Commutative cocycles and stable bundles over surfaces

Abstract

Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind, and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles, and are analyzed via the point-wise inversion operation on commutative cocycles. 13 4.

On the second homotopy group of the classifying space for commutativity in Lie groups

“…To a G-valued cocycle α over C one can associate a G-bundle E α → S 2 . By [14,Lemma 3.3] if α classifies E α . Moreover, by [14, Lemma 3.2] E α is clutched by the map ϕ α : S 1 → G given by…”

“…The space B(2, G) sitting in the first term of this filtration arises by assembling together the spaces of commuting tuples in G. The homotopy fiber of the inclusion B(2, G) ֒→ BG, denoted E(2, G) can be thought of as the difference between B(2, G) and BG, and in some sense measures how far is G from being abelian. In this note we study the second homotopy group π 2 (E(2, G)) for a compact Lie group G. The homotopy classes in π 2 (E(2, G)) for the orthogonal matrix groups G = O(n), were previously used in [14] to produce non-standard classes in the reduced commutative orthogonal K-theory of closed connected surfaces. This cohomology theory is a variant of orthogonal K-theory (as defined in [3] and further studied in [4]), whose classes are represented by orthogonal vector bundles equipped with transition functions that point-wise commute.…”

“…Let [G, G] denote the commutator subgroup of G. To prove our main result we combine the commutator map c : E(2, G) → B[G, G] introduced in [6] with the techniques developed in [14] of producing new classifying maps from old ones by inverting cocycles that point-wise commute. For instance, in the case of the special orthogonal groups SO(n), this procedure can be translated into c * : π 2 (E(2, SO(n))) → π 2 (BSO(n)) being an isomorphism, for every n ≥ 3.…”

Higher Generation by Abelian Subgroups in Lie Groups

“…Because the inclusion SO(2) → O(2) induces an isomorphism on π 2 of the classifying spaces, it is equivalent to showing, using Remark 11, that the map iφ −1 p : E(2, O(2)) → BO(2) is surjective on π 2 . Surjectivity follows from results in[17] as we will now explain. The authors construct a map f 1 :S 2 → B(2, O(2)) such that if 1 is null-homotopic but iφ −1 f 1 is a generator of π 2 (BO(2)) ∼ = Z ([17, Prop.…”

“…Surjectivity follows from results in[17] as we will now explain. The authors construct a map f 1 :S 2 → B(2, O(2)) such that if 1 is null-homotopic but iφ −1 f 1 is a generator of π 2 (BO(2)) ∼ = Z ([17, Prop. 3.5]).…”