There is a product decomposition of a compact connected Lie group G at the prime p, called the mod p decomposition, when G has no p-torsion in homology. Then in studying the multiplicative structure of the p-localization of G, the Samelson products of the factor space inclusions of the mod p decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the p-localized exceptional Lie groups when the factor spaces are of rank ≤ 2, that is, G is quasi-p-regular.

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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.

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Abstract. For a simple n-polytope P , a quasitoric manifold over P is a 2n-dimensional smooth manifold with a locally standard action of the n-dimensional torus for which the orbit space is identified with P . This paper shows the topological classification of quasitoric manifolds over the dual cyclic polytope C n (m) * , when n > 3 or m − n = 3. Besides, we classify small covers, the "real version" of quasitoric manifolds, over all dual cyclic polytopes.