Study of coupled K-theory and its applications to non-linear actions

因此，手术障碍被定义为二次模M=(K_k(X；Z)，λ，μ)或具有位置映射a：〓_o→K_k(X；Z)的某些等价类。这里，λ和μ分别是交集形式和自交形式。在这项研究中，我们发展了一个由Z[G]-二次模M、Z_2[G]-二次模M_2、定位图a：〓→K_k(X；Z)和定位图a：〓_2→K_k(X；Z)组成的四(M，M_2，a，a_2)耦合K-理论。Z_2)。我们构造了新的等变手术障碍，定义了新的拉格朗日和代谢形式，对它们和二次模进行了分类，从几何的角度研究了手术技巧，并将这些集合在一起，构造了新的等变手术理论。我们定义了耦合厄米特模和一个新的特殊Grothendieck-Witt群。进一步，我们研究了关于Mackey函子、Burnside环上的模和Green函子的群，结合上面的结果和Oliver的结果，我们得到了圆盘和球面上的各种非线性光滑作用量。特别地，对于幂零Oliver群和完全群，我们确定了球面上光滑作用的单连通不动点流形。

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.