| Co-Investigator(Kenkyū-buntansha) |
IKEHATA Shuichi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (20116429)
NAKAJIMA Atsushi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30032824)
SHIMAKAWA Kazuhisa Okayama University, Faculty of Science Professor, 理学部, 教授 (70109081)
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| Research Abstract |
So fer, surgery obstructions have been defined as certain equivalence classes of quadratic modules M=(K_k(X;Z),λ,μ) or ones with positioning maps a: 〓_o→ K_k(X;Z). Here λ and μ are the intersection form and the selfintersection form, respectively. In this research, we developed a new theory, namely a coupled K-theory, of quadruple (M, M_2, a, a_2) consisting of a Z[G]-quadratic module M, a Z_2[G]-quadratic module M_2, a positioning map a: 〓→ K_k(X;Z), and a positioning map a: 〓_2→ K_k(X;Z_2).
We constructed new equivariant surgery obstructions, define new Lagrangians and metabolic forms, classified them and quadratic modules, studied surgery technique from the view point of geometry, and putting all this together, constructed a new equivariant surgery theory.
We defined coupled Hermitian modules and a new special Grothendieck-Witt group. Furthermore, we studied the group in the respect of a Mackey functor, a module over the Burnside ring, and a Green functor.
Combining the results above with Oliver's results, we obtained various nonlinear smooth actions on disks and spheres. In particular, for nilpotent Oliver groups and perfect groups, we determined simply connected fixed point manifolds of smooth actions on spheres.
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