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Abstract: Let be either the real number field or the complex number field and the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given -vector bundle over to be stably extendible to for every . In this paper, we simplify the theorems and apply them to the tangent bundle of , its complexification, the normal bundle associated to an immersion of in , and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5]. [Copyright &y& Elsevier]

Abstract: We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid with G compact. [Copyright &y& Elsevier]

Abstract: Let denote the -dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over to be stably extendible to for every , and establish the formula on the power (k-fold) of a real vector bundle ζ over . Moreover, we answer the stable splitting problem for real vector bundles over by means of arithmetic conditions. [Copyright &y& Elsevier]

Abstract: Let F be the real number field R or the complex number field C, and let denote the real projective n-space. In this paper, we study the conditions for a given F-vector bundle over to be stably extendible to for every , and establish the formulas on the power (r-fold) of an F-vector bundle ζ over . Our results are improvements of the previous papers [T. Kobayashi, H. Yamasaki, T. Yoshida, The power of the tangent bundle of the real projective space, its complexification and extendibility, Proc. Amer. Math. Soc. 134 (2005) 303–310] and [Y. Hemmi, T. Kobayashi, Min Lwin Oo, The power of the normal bundle associated to an immersion of , its complexification and extendibility, Hiroshima Math. J. 37 (2007) 101–109]. Furthermore, we answer the stable splitting problem for F-vector bundles over by means of arithmetic conditions. [Copyright &y& Elsevier]

Abstract: It is a classical theorem of Milnor that for every vector bundle over , all the Stiefel–Whitney classes vanish if and only if . We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel–Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces. [Copyright &y& Elsevier]

Abstract: Let p be a prime number. It is proven that for the product of free actions of on modp homology spheres , , where s are assumed to be odd if p is odd, for any continuous map the set has dimension at least , provided for all i (). [Copyright &y& Elsevier]