(I) Equivariant degree theory: Definition, properties and computations

A topological degree, in its simplest form, may be thought as a generalization of the {\it winding number} of a continuous circle map, which counts how many times the image of the map has traveled counterclockwise around the origin. This count remains unchanged if the map is perturbed slightly. Also, the addition of winding numbers corresponds to the conjunction of maps, and the negation of winding numbers can be realized by rewinding the direction of maps. The topological degree is thus usually referred as ``an algebraic count of the zeros of a continuous map''.

Equivariant degree theory is a topological degree theory that is concerned with {\it equivariant maps}, that is, maps that commute with the actions of a group on their space of domain and image. A main objective of the equivariant degree theory is to attain the topological structure of the zeros of an equivariant map and their algebraic properties induced by the equivariance.

In the first part of this course, we give a definition of equivariant degree, using a list of its most important properties such as existence and homotopy invariance. We also give examples of computations and among others, computational formulas for basic maps, which will be used frequently in application.