2 Pages

Introduction to Part I

WithMarvin J. Greenberg, John R. Harper

The wellspring of ideas leading to algebraic topology was the perception, developed largely in the latter half of the nineteenth century, that many properties of functions were invariant under “deformations.” For example, Cauchy’s theorem and the calculus of residues in complex analysis assert invariance of complex integrals with respect to continuous deformations of curves. Perhaps the true starting point was Riemann’s theory of abelian integrals. It was here that the significance of the connectivity of surfaces was recognized. The interested reader is strongly encouraged to examine Felix Klein’s exposition of Riemann’s theory [80], during the study of algebraic topology.