Infiniteness ofA_∞-types of gauge groups

Abstract

Abstract. Let G be a compact connected Lie group and let P be a principal G-bundle over K. The gauge group of P is the topological group of automorphisms of P . For fixed G and K, consider all principal G-bundles P over K. It is proved in [CS, Ts] that the number of A n -types of the gauge groups of P is finite if n < ∞ and K is a finite complex. We show that the number of A ∞ -types of the gauge groups of P is infinite if K is a sphere and there are infinitely many P .

The mod-p homology of the classifying spaces of certain gauge groups

“…Under the assumption of Theorem 1.1 the Lie group G is p-locally homotopy equivalent to the product l i=1 S 2n i −1 , so we can also calculate the Poincaré series of H * (BG k ) by Theorem 1.1 (Corollary 4.5). Remarkably, Theorem 1.1 implies that the mod-p homology of BG k is independent of k for p large, whereas there is more than one p-local homotopy type in the family {BG k } k∈Z for p large as was proved in [KT1]. The approach to Theorem 1.1 is to consider the Serre spectral sequence applied to the fiberwise coproduct of (1.2).…”

“…The difference of the mod- homology of for in Theorems 1.1 and 1.2 comes from the homotopy commutativity of ; it is homotopy commutative if and only if as in [17]. This is notable because the product decomposition as -spaces is guaranteed by the higher homotopy commutativity of as in [11, 12], whereas theorem 1.1 shows the homological product decomposition as -spaces.…”

On the homotopy types of Sp(n) gauge groups

“…The first classification was done by the second named author [17] for G = SU (2), and since then, considerable effort has been made for the classification when G is of law rank [7,8,9,10,15,17,23,24,25]. Properties of gauge groups related with the classification of the homotopy types have also been intensively studied [3,11,13,14,16,22].…”