On Topology: On Homotopy (- 1999)

"A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector. A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent  coincides with its end point and tangent." (Steven Smale,"Regular Curves on Riemann Manifolds", 1956)

"Of course, there are a number of important problems left in differential topology that do not reduce in any sense to homotopy theory and topologists can never rest until these are settled. But, on the other hand, it seems that differential topology has reached such a satisfactory stage that, for it to continue its exciting pace, it must look toward the problems of analysis, the sources that led Poincare to its early development." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"One of the most basic ideas used in the study of mappings from one space to another is that of homotopy. Two mappings are said to be homotopic if one can be 'deformed' into the other through a one-parameter family of mappings between the same spaces. Sometimes further conditions are imposed on the family of mappings [...]" (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology." (Brayton Gray, "Homotopy Theory", Pure and Applied Mathematics Vol. 64, 1975)

"A regular homotopy of an immersion is a deformation through immersions for which the matrix of first partials itself varies smoothly." (George K Francis, "A Topological Picturebook", 1987)

"The story of everting spheres in 3-space by regular homotopies is the case history of a nontrivial visualization problem of remarkable complexity and compelling beauty. The task is to show the motion of a spherical surface through itself in space so that, without tearing or creasing, the surface is turned inside out. That so many different graphical methods were applied to the same problem, when it is more in the nature of mathematics to display the versatility of one method by applying it to a variety of different problems, makes it a paradigm for descriptive topology." (George K Francis, "A Topological Picturebook", 1987)