Allen Hatcher - Collected Quotes

"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)

"Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images - the ‘lanterns’ of algebraic topology, one might say - are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images." (Allen Hatcher, "Algebraic Topology", 2001)

"Algebraic topology is most often concerned with properties of spaces that depend only on homotopy type, so local topological properties do not play much of a role." (Allen Hatcher, "Algebraic Topology", 2001)

"An interesting feature of homology that begins to emerge after one has worked with it for a while is that it is the basic properties of homology that are used most often, and not the actual definition itself. This suggests that an axiomatic approach to homology might be possible. This is indeed the case [...]. One could take the viewpoint that these rather algebraic axioms are all that really matters about homology groups, that the geometry involved in the definition of homology is secondary, needed only to show that the axiomatic theory is not vacuous. The extent to which one adopts this viewpoint is a matter of taste, and the route taken here of postponing the axioms until the theory is well-established is just one of several possible approaches." (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)

"Since all groups can be realized as fundamental groups of spaces, this opens the way for using topology to study algebraic properties of groups." (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)