Let’s get acquainted with mapping degree!

Abstract

Given a continuous map f: M → N between oriented manifolds of the same dimension, the associated degree deg(f) is an integer which evaluates the number of times the domain manifold M “wraps around” the range manifold N under the mapping f. The mapping degree is met at almost every corner of mathematics. Some of its avatars, pseudonyms, or close relatives are “winding number”, “index of a vector field”, “multiplicity of a zero”, “Milnor number of a singularity”, “degree of a variety”, “incidence numbers of cells in a CW-complex”, etc. We review some examples and applications involving this important invariant. One of emerging guiding principles, useful for a mathematical student or teacher, is that the study of mathematical concepts which transcend the boundaries between different mathematical disciplines should receive a special attention in mathematical (self)education.