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

为此，将手术障碍物定义为二次模M=（K_k（X;Z），λ，μ）或具有定位映射a：λ_o→ K_k（X;Z）的二次模的等价类。这里λ和μ分别是相交形式和自相交形式。在这项研究中，我们发展了一个新的理论，即耦合K理论，（M，M_2，a，a_2）由Z[G]-二次模M、Z_2[G]-二次模M_2、定位映射a：k → K_k（X;Z）和定位映射a：k → K_k（X; Z_2），构造了新的等变手术障碍物，定义了新的拉格朗日量和代谢形式，并对它们和二次模进行了分类，从几何的角度研究了手术技术，在此基础上，我们定义了耦合Hermitian模和一个新的特殊Grothendieck-Witt群，并给出了一个新的等变外科理论。进一步，我们从Mackey函子、伯恩赛德环上的模和绿色函子的角度研究了该群，并将上述结果与奥利弗的结果相结合，得到了圆盘和球面上的各种非线性光滑作用.特别地，对于幂零奥利弗群和完全群，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.