Abstract

Given a continuous map $f : M\rightarrow N$ between oriented manifolds of the same dimension, the associated {ıt 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.

Keywords: Mapping degree; winding number.

Pages:  119$-$136     

Volume  XIV ,  Issue  2 ,  2011