"A space having the homotopy type of a point is called contractible. This amounts to requiring that the identity map of the space be nullhomotopic, that is, homotopic to a constant map. In general, this is slightly weaker than saying the space deformation retracts to a point; [...]" (Allen Hatcher, "Algebraic Topology", 2001)
"From the viewpoint of homotopy theory, cohomology is in some ways more basic than homology. [...] cohomology has a description in terms of homotopy classes of maps that is very similar to, and in a certain sense dual to, the definition of homotopy groups." (Allen Hatcher, "Algebraic Topology", 2001)
"Homotopy theory begins with the homotopy groups πn(X), which are the natural higher-dimensional analogs of the fundamental group. These higher homotopy groups have certain formal similarities with homology groups. For example, πn(X) turns out to be always abelian for n ≥ 2, and there are relative homotopy groups fitting into a long exact sequence just like the long exact sequence of homology groups. However, the higher homotopy groups are much harder to compute than either homology groups or the fundamental group, due to the fact that neither the excision property for homology nor van Kampen’s theorem for π1 holds for higher homotopy groups." (Allen Hatcher, "Algebraic Topology", 2001)
"The computability of homology groups does not come for free, unfortunately. The definition of homology groups is decidedly less transparent than the definition of homotopy groups, and once one gets beyond the definition, there is a certain amount of technical machinery to be set up before any real calculations and applications can be given." (Allen Hatcher, "Algebraic Topology", 2001)
"The operation of collapsing a subspace to a point usually has a drastic effect on homotopy type, but one might hope that if the subspace being collapsed already has the homotopy type of a point, then collapsing it to a point might not change the homotopy type of the whole space." (Allen Hatcher, "Algebraic Topology", 2001)
"A main problem of topology is the classification of topological spaces: Given two spaces X and Y, are they homeomorphic? This is usually a very difficult question to answer without employing some fairly sophisticated machinery, and the idea of algebric topology is that in which one should transform such topological problems into algebraic problems in order to have a better chance of solution. It turns out, however, that the algebric techniques are usually not delicate enough to classify spaces up to homeomorphism. Hence we shall introduce the notion of homotopy, in order to achieve a somewhat coarser classification." (D Chatterjee, "Topology: General & Algebraic", 2003)
"The concept of path-connectedness, in which it is required that it be possible to reach any point in the space from any other point along a continuous path is necessary for the notion of fundamental group. This approach is especially useful in studying connectivity properties from an algebraic point of view, e.g., via homotopy theory." (D Chatterjee, "Topology: General & Algebraic", 2003)
"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)
"One might ask why the idea of homotopy equivalence between spaces is worth studying. One of the main reasons is that it gives a powerful tool to determine the fundamental group of a space." ("Introduction to Topology", 2007)
"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-hour-long 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)
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