Algebraic Topology

In General
* Idea: The branch of mathematics that studies intrinsic aspects of topological spaces, using algebraic concepts such as groups and rings; It expresses a remarkable interpenetration of algebra and topology.
* Goal: In general terms, we would like to get as close as possible to classifying topological spaces, i.e., introducing enough criteria to distinguish equivalence classes of topological spaces; This has been achieved fully only for restricted classes of topological spaces, e.g., 2D closed surfaces; These criteria are given by assigning to spaces some topological invariants, like homotopy, homology, isotopy (more difficult and restrictive), dimension, connectedness, compactness, etc; Most of the useful topological invariants are homotopy invariants.
* Goal, more precisely: We want to construct a series of functors from the category Top to the category of groups, that allow to translate statements about topological spaces into statements about groups, and solve topological questions by algebraic methods; Typical problems are the homeomorphism, lifting, and retraction problems.
* History: Before the 1960's, it was "passive" and concentrated on invariants of known spaces; Also, classically it was the local study of geometrical objects; Later it became "activist" and started constructing new spaces or structures, sometimes using "control spaces" (surgery); Now the global structure viewpoint dominates, but global theorems are also applied to local properties.
@ References: Poincaré JEP(1895); & Eilenberg, MacLane.

Isotopy Theory
$ Diffeomorphisms: Two diffeomorphisms are isotopic if they are connected in the space Diff, with C^infty topology.
$ Homeomorphisms: A homeomorphism f : X → X is isotopic to the identity if there is a homotopy F: X × I → X, f(t): X → X a homeomorphism for all t, with f(0) = id, f(1) = f.

Other Branches > s.a. cohomology; Combinatorial Topology; homology; homotopy.
* Surgery: The most famous activist theory in algebraic topology; Developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others, a method for comparing homotopy types of topological spaces with diffeomorphism or homeomorphism types of manifolds of dimension n 5; One begins with a space X with global algebraic properties of a manifold (> see Poincaré Duality) and tries to construct a manifold M and a homotopy equivalence M → X (...).
@ Surgery: Madsen & Milgram 78; Kreck AM(99)m.GT [generalization].

Specific Concepts > see Betti Numbers; differential forms.

References
@ Books, II: Croom 78.
@ Books, III: Spanier 66.
@ Books: Lefschetz 42; Eilenberg & Steenrod 52 [axioms]; Wallace 57, 63; Godement 58; Bourgin 63; Greenberg 67; in Singer & Thorpe 67; Artin & Braun 69; Adams 72; Dold 72; Maunder 72; Lefschetz 75; Switzer 75; Agoston 76; Massey 77; Greenberg & Harper 81; Munkres 84; Pontrjagin 86; Madsen & Tornehave 97.
@ Books for physicists: Nash & Sen 83; Nakahara 03.
@ Based on knots: Przytycki 01.
@ Local algebraic topology: Quinn NAMS(86).

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 14 jun 2008