The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S 2d for 2 d p − 1 and n 2p − 1, we give a concrete classification of their p-local homotopy types.
The first aim of this paper is to study the p-local higher homotopy commutativity of Lie groups in the sense of Sugawara. The second aim is to apply this result to the p-local higher homotopy commutativity of gauge groups. Although the higher homotopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application. The third aim is to resolve the 5-local higher homotopy non-commutativity problem of the exceptional Lie group G 2 , which has been open for a long time.
Abstract. The equivalence class of a principal G 2 -bundle over S 4 is classified by the value k ∈ Z of the second chern class. In this paper we consider the homotopy types of the corresponding gauge groups G k , and determine the number of homotopy types up to one factor of 2.
Given an action of a finite group G, we can define its index. The G-index roughly measures a size of the given G-space. We explore connections between the Gindex theory and topological dynamics. For a fixed-point free dynamical system, we study the Z p -index of the set of p-periodic points. We find that its growth is at most linear in p. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years.
Abstract.A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n) (p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.2010 Mathematics Subject Classification. 55Q15.
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