Study of equivariant homology surgery theory and its applications

(1)设G为有限群，Y为光滑G作用的同调盘，f: X→Y为G框架映射，H为G的子群，设Y有一个G不动点和一个各向同性子群H的点，构造了G连通和f#_G G x_H D(f): X #_G G x_H D(X)→Y。证明了广义群与原群同构。利用这个新概念，我们证明了上述g连通和的手术障碍求和公式。(3)对于Z_{(p)}，当p是一个素数，使得Y的基群的阶相同时，证明了等变同调阻塞群与Wall群同构。因此，我们可以建立等变同调手术梗阻群的诱导-限制理论。(4)利用上述结果，证明了间隙奥利弗群G的G不动点集的一个消插定理。(5)在G为幂零奥利弗群或非平凡完美群的球面上，确定了光滑G作用的G不动点集的流形。

(1)Let G be a finite group, Y a homology disk with a smooth G-action, f : X → Y a G-framed map, H a subgroup of G. Suppose that Y has a G-fixed point and a point of isotropy subgroup H. Then we constructed G-connected sum f#_G G x_H D(f) : X #_G G x_H D(X) → Y.(2)We generalized the notion of quadratic forms of Cappell-Shaneson and their group. We proved the generalized group is isomorphic to the original one. Using this new notion, we proved a sum formula of surgery obstructions for the G-connected sum above.(3)For Z_{(p)} where p is a prime such that the order of fundamental group of Y, we proved that the equivariant homology obstruction group is isomorphic to the Wall group. So, we could develop induction-restriction theory of equivariant homology surgery obstruction groups.(4)Using above results, we proved a deleting-inserting theorem of G-fixed point sets for gap Oliver group G.(5)We decided manifolds appearing as the G-fixed point sets of smooth G-actions on spheres where G was a nilpotent Oliver group or a nontrivial perfect group.