2001 Fiscal Year Final Research Report Summary
Homotopy theoretical research of manifolds
Project/Area Number 
11640069

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Kanazawa University 
Principal Investigator 
ISHIMOTO Hiroyasu Kanazawa Univ., Fac. Science, Prof. > 金沢大学, 理学部, 教授 (90019472)

CoInvestigator(Kenkyūbuntansha) 
TOMARI Masataka Kanazawa Univ., Fac. Science, Assoc. Prof., 理学部, 助教授 (60183878)
MORISHITA Masanori Kanazawa Univ., Fac. Science, Assoc. Prof., 理学部, 助教授 (40242515)
SUGANO Takashi Kanazawa Univ., Fac. Science, Prof., 理学部, 教授 (30183841)
IWASE Zunici Kanazawa Univ., Fac. Science, Research Assoc., 理学部, 助手 (70183746)
FUJIOKA Atsushi Kanazawa Univ., Fac. Science, Assoc. Prof., 理学部, 講師 (30293335)

Project Period (FY) 
1999 – 2001

Keywords  primary manifolds / homotopy equivalent manifolds / constant mean curvature surfaces / harmonic inverse mean curvature surfaces / hypersurface isolated singularities / Milnor number / Pn'mes and Knots / automorphic forms 
Research Abstract 
(1) Ishimoto studied the problem whether the matter corresponding to the Poincare conjecture holds or not for primary manifolds which are mspheres with attached qhandles in the metastable range. For that purpose, he intended to extend the JamesWhitehead theorem to the one for primary manifolds and succeeded in such case that the quadratic forms which distinguish primary manifolds take values in a cyclic group. Using the result, he proved in almost all cases that the matter in question also valid when the cyclic group is Z_<24>, adding to the results already obtained. (2) Fujioka studied the fundamental properties of harmonic inverse mean curvature surfaces which are natural generalization of constant mean curvature surfaces. In particular, he characterized such a surface as the one which admits a transformation preserving a certain quantity represented with curvature. He studied also the Bonnet surfaces. (3) Tomari studied the theory of multiplicity of filtered rings, and as an application, he constructed a criterion formula for the Milnor number of f which gives the definition of hyper surface isolated singularities, using the weight of coordinates and the Taylor expansion. (4) Morishita studied analogies between knots and primes, 3manifolds and number fields, basing on the analogy between link groups and Galois groups, and tried to bridge between the algebraic number theory and the 3dimensional topology. He also studied with K, Murasugi in Toronto. (5) Sugano studied the automorphic forms on unitary groups of degree 3 in number theory. He gave the explicit expansion for Eisenstein series and Kudla lift images using primitive theta functions, and gave the nonvanishing condition for the Kudla lift in terms of the periods.

Research Products
(12 results)