55P: Homotopy theory
This is the study of topological spaces using a broader equivalence than homeomorphism. It's really the natural setting for algebraic topology because the traditional algebraic invariants (fundamental group, etc.) are isomorphic not only for homeomorphic spaces but for spaces of the same homotopy type. Since most of the information about these invariants is already on the algebraic topology page, there isn't much left here. There are some comments about contractibility.
Rather narrow for our purposes but a place to start: Moore, J. C.; Neisendorfer, J. A.: "A view of some aspects of unstable homotopy theory since 1950", in Homotopy theory -- Durham 1985, 117--148, London Math. Soc. Lecture Note Ser., 117, Cambridge Univ. Press, Cambridge-New York, 1987. MR89e:55002
For simple homotopy type, see 57Q10
Parent field: 55: Algebraic Topology
Browse all (old) classifications for this area at the AMS.
Recent texts have tended to be more specialized. Appropriate for all of section 55P is the book of Whitehead, George W.: "Elements of homotopy theory", GTM 61, Springer-Verlag, New York-Berlin, 1978. 744 pp. ISBN 0-387-90336-4 MR80b:55001