拓扑手术中$ρ$ -不变量与周期性的可加性
The additivity of the $ρ$-invariant and periodicity in topological surgery
摘要：
对于dim（M）>= 5的闭拓扑流形M，拓扑结构集S（M）有一个阿贝尔群结构，它可以与Ranicki定义的M的代数结构群等同.若dim（M）= 2d-1，M是定向的，且M具有到有限群G的分类空间的映射，则约化rho-不变量定义了一个函数\wrho：S（M）\to \QQ R_{hat G} ^{（-1）^d}到G的复表示环的某个子商.证明了当2d-1 >= 5时，函数\wrho是同态.一路上，我们给出了一个详细的证明，几何定义的地图，由于Cappell和Weinberger实现了8倍Siebenmann周期图的拓扑手术。
英文摘要：
For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced rho-invariant defines a function, \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of the complex representation ring of G. We show that the function \wrho is a homomorphism when 2d-1 >= 5. Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8-fold Siebenmann periodicity map in topological surgery.
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