Ivancevic_Applied-Diff-Geom

350 Applied Differential Geometry: A Modern IntroductionU ⊂ M C . Recall (see Introduction) that γ is homotopic to η if there existsa C 0 −function ψ : [a, b] × [c, d] → U defined on a rectangle [a, b] × [c, d] ⊂U, such that ψ(t, c) = γ(t) and ψ(t, d) = η(t) for all t ∈ [a, b]. Foreach number s ∈ [c, d] we may view the function |psi s (t) = ψ(t, s) as acontinuous curve defined on [a, b], and we may view the family of continuouscurves ψ s as a deformation of the path γ to the path η. It is said that thehomotopy ψ leaves the end points fixed if we have ψ(a, s) = γ(a) andψ(b, s) = γ(b) for all values of s ∈ [c, d]. Similarly, when we speak of ahomotopy of closed paths, we assume that each path ψ s is a closed path.Let γ, η be paths in an open set U ⊂ M C having the same beginning andend points. Assume that they are homotopic in U. Let f be holomorphicon U. Then ∫ γ f = ∫ f. The same holds for closed homotopic paths in U.ηIn particular, if γ is homotopic to a point in U, then ∫ f = 0. Also, it isγsaid that an open set U ⊂ M C is simply–connected if it is connected and ifevery closed path in U is homotopic to a point.In the previous example we found that12πI∫γ1dz = 1,zif γ is a circle around the origin, oriented counterclockwise. Now we definefor any closed path γ its winding number with respect to a point α to beW (γ, α) = 1 ∫12πi z − α dz,provided the path does not pass through α. If γ is a closed path, thenW (γ, α) is an integer.A closed path γ ∈ U ⊂ M C is homologous to 0 in U if∫1dz = 0,z − αγfor every point α not in U, or in other words, W (γ, α) = 0 for every suchpoint.Similarly, let γ, η be closed paths in an open set U ⊂ M C . We say thatthey are homologous in U, and write γ ∼ η, if W (γ, α) = W (η, α) for everypoint α in the complement of U. We say that γ is homologous to 0 in U,and write γ ∼ 0, if W (γ, α) = 0 for every point α in the complement of U.If γ and η are closed paths in U and are homotopic, then they arehomologous. If γ and η are closed paths in U and are close together, thenthey are homologous.γ