Surgery of involutions with middle-dimensional fixed point set
Let X and Y be simply-connected closed manifolds with smooth \({\mathbb{Z}}_ 2\)-actions and let f: \(X\to Y\) be a degree one equivariant map. The authors study the problem of finding a \({\mathbb{Z}}_ 2\)-bordism of f to f': X'\(\to Y\) which is a (not necessarily equivariant) homotopy equivalence. There are several technical assumptions made in order to make \({\mathbb{Z}}_ 2\)-equivariant surgery possible. The special hypothesis emphasized in this paper in addition to others is that ``the dimension of the fixed point set is at most (1/2)dim X and there is exactly one component of the fixed set whose dimension is equal to (1/2)dim X''. The paper is very much in the spirit of \textit{K. H. Dovermann}'s [Mich. Math. J. 28, 267-287 (1981; Zbl 0491.57014)]. The main results of the paper give necessary and sufficient conditions for the vanishing of the surgery obstructions in terms of signatures and Arf invariants. The techniques are adapted from standard surgery theory.
scientific article; zbMATH DE number 4071941