57: Manifolds and cell complexes
Manifolds are spaces like the sphere which look locally like Euclidean space. In particular, these are the spaces in which we can discuss (locally-)linear maps, and the spaces in which to discuss smoothness. They include familiar surfaces. Cell complexes are spaces made of pieces which are part of Euclidean space, generalizing polyhedra. These types of spaces admit very precise answers to questions about existence of maps and embeddings; they are particularly amenable to calculations in algebraic topology; they allow a careful distinction of various notions of equivalence. These are the most classic spaces on which groups of transformations act. This is also the setting for knot theory.
Thorough, excellent coverage is provided by Dieudonné, Jean, "A history of algebraic and differential topology. 1900--1960", Birkhäuser Boston, Inc., Boston, MA, 1989, 648 pp. ISBN 0-8176-3388-X MR90g:01029. He has a similar (shorter!) survey done a little later (see MR95k:55001)
Perhaps it is easiest to use classic literature to understand differential topology: Flatland; here are two Backup sites and the home page for Project Gutenberg.
There are two other topology pages:
Algebraic topology -- definitions and computations of fundamental groups, homotopy groups, homology and cohomology. This includes Homotopy theory -- studies of spaces in the homotopy category (but without applications to questions on algebraic invariants)
General topology is for spaces without the local Euclidean nature of the spaces in this section. There is however some semblance to the present topic among metric spaces.
Simplicial complexes are essentially polyhedra.
Problems specific to Euclidean space may be treated in the geometry pages
Questions regarding, say, smoothness on balls and spheres are in the FAQ for spheres.
For complex manifolds, See 32C10
Geometric topology is a natural language in which to study families of motions; applications include some topics in mechanics of moving particles and systems.
Browse all (old) classifications for this area at the AMS.
Bourbaki, N., "Variétés différentielles et analytiques" Hermann, Paris 1971 99 pp.
Fox, R. H.: "A quick trip through knot theory", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 120--167; Prentice-Hall, Englewood Cliffs, N.J. 1962 MR25#3522
There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period.
Cannon, J. W.: "The recognition problem: what is a topological manifold?", Bull. Amer. Math. Soc. 84 (1978), no. 5, 832--866. MR58#13043