On Homotopy

"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)

"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)

"Homotopy theory came into existence in the 1930s, after Hopf’s introduction of the fibrations that now bear his name and Hurewicz’s introduction of the higher homotopy groups together with some of their fundamental  properties. From this point on, homotopy theory interacted strongly with the other tools of algebraic topology, e.g. homology theory, cohomology theory, spectral sequences, it moved slowly to the forefront of algebraic topology in general, led to new synthesis in the form of homotopical algebra and is now being applied in a wide variety of fields [...]" (Jean-Pierre Marquis, "A Path to the Epistemology of Mathematics: Homotopy Theory", [in J Ferreirós & J J Gray (Eds.), "The Architecture of Modern Mathematics: Essays in History and Philosophy"]  2006)

"Two continuous mappings are contained in the same mapping class iff they can be continuously deformed into each other. In important special cases, the space of mapping classes can be equipped with an additional group structure. This leads to Poincaré’s fundamental group and the higher homotopy groups of topological spaces." Eberhard Zeidler "Quantum Field Theory II: Quantum Electrodynamics", 2006)

"Topology is a child of twentieth century mathematical thinking. It allows us to consider the shape and structure of an object without being wedded to its size or to the distances between its component parts. Knot theory, homotopy theory, homology theory, and shape theory are all part of basic topology. It is often quipped that a topologist does not know the difference between his coffee cup and his donut - because each has the same abstract 'shape' without looking at all alike." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"The notion of homotopy formalizes the idea of continuously altering (or 'wiggling' ) maps of topological spaces. [...] The intuition is that a homotopy between two maps f and g is a one-hourlong movie, which starts showing the map f , and ends with the map g." (Renzo Cavalieri and Eric Miles, "Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory", 2016)