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）或具有定位映射a：〓_o→K_k（X；Z）的等价类。这里λ和μ分别是交点形式和自交点形式。本文提出了由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：〓→K_k（X;Z_2）组成的四重元（M, m2, a, a_2）的耦合k理论。我们构造了新的等变手术障碍，定义了新的拉格朗日量和代谢形式，并对它们进行了分类和二次模，从几何的角度研究了手术技术，将这些综合起来，构造了新的等变手术理论。我们定义了耦合的厄米模和一个新的特殊的Grothendieck-Witt群。进一步研究了Mackey函子、Burnside环上的模和Green函子的群。将上述结果与奥利弗的结果相结合，我们得到了圆盘和球体上的各种非线性光滑作用。特别地，对于幂零奥利弗群和完美群，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.