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A complete invariant defined for (closed, connected, orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself is reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In previous papers by the first author, a characteristic lattice point invariant and a characteristic rational invariant defined for the 3-manifolds were constructed which also produced a smooth real function with the definition interval ( − 1 , 1 ) as a characteristic invariant defined for the 3-manifolds. In this paper, a complex number-valued characteristic invariant for the 3-manifolds whose norm is smaller than or equal to one half is introduced by using an embedding of a set of lattice points called the ADelta set, distinct from the PDelta set, into the set of complex numbers. The distributive situation for the invariants of the 3-manifolds of lengths up to 10 is plotted in the complex number plane with radius smaller than or equal to one half. By using this complex number-valued characteristic invariant, a holomorphic function with the unit open disk as the definition domain which is called the characteristic quantity function is constructed as a characteristic invariant defined for the 3-manifolds. [ABSTRACT FROM AUTHOR]

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: For a surface F, the Kauffman bracket skein module of , denoted , admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have , a vector space over . In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space has five distinct traces. One trace, the Yang–Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on correspond to the four singular points of the moduli space of flat -connections on the torus. [Copyright &y& Elsevier]