The Topology of a Group Action

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.

A characterization of unbounded generalized meromorphic operators

In this paper, we study the class of unbounded generalized meromorphic operators GM(E,?), where E is a finite subset of C, which generalizes the notion of unbounded meromorphic operators. More precisely, we give a decomposition and some characterizing properties of these operators based on the punctured neighborhood theorem and the operational calculus for unbounded operators.