Degree Theory: Old and New

Abstract

The theory of degree has a long history which is a cascade of successive generalizations. Presumably, the oldest notion is the degree of a (smooth) map u from S ¹ into S ¹ (S ¹ = the unit circle). The degree of u, also called winding number, counts “how many times u covers its range taking into account the algebraic multiplicity.” More generally, a smooth (say C ¹) map u from S ¹ into ℂ, such that u ≠ 0 on S ¹ has a degree which may be computed through the very classical integral formula

$$\deg u = \frac{1} {{2\pi i}}\int_{S^1 } {\frac{{\dot u}} {u}}$$

((1.1))

which measures the “algebraic change of phase” of u as the variable goes around S ¹ once. Similarly, if Γ is a simple curve in ℝ² and u is a smooth map from Γ into S ¹, then its degree can be computed as

$$\deg (u,\Gamma ) = \frac{1} {{2\pi }}\int_\Gamma {u \times u_\tau ,}$$

((1.2))

where × denotes the cross product of vectors in ℝ² (here S ¹ is viewed as a subset of ℝ², not ℂ)and u _τ denotes the tangential derivative of u along Γ.