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 $$\cal G$$ 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 1960s, 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.
@ History: Poincaré JEP(1895); & Eilenberg, MacLane; Dieudonné 09.

Isotopy Theory
$Diffeomorphisms: Two diffeomorphisms are isotopic if they are connected in the space Diff, with C topology.$ Homeomorphisms: A homeomorphism f : XX is isotopic to the identity if there is a homotopy F: X × I → X, f(t): XX 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 MX (...).
@ Surgery: Madsen & Milgram 78; Kreck AM(99)m.GT [generalization]; in Wall 16; > s.a. Dehn Surgery.

Specific Concepts > see Betti Numbers; differential forms; exact sequence.

References
@ Books, II: Croom 78; Adhikari 16.
@ Books, III: Spanier 66; Hatcher 01; Davis & Kirk 01.
@ 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; Dodson & Parker 97 [and applications]; Kreck 10 [differential algebraic topology]; Borisovich et al 11.
@ Books for physicists: Nash & Sen 83; Nakahara 03; Robins a1304-ch [introduction, including computational work].
@ Based on knots: Przytycki 14.
@ Local algebraic topology: Quinn NAMS(86); > s.a. homology.