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.

> __Online resources__:
see Wikipedia page.

**Isotopy Theory**

$ __Diffeomorphisms__: Two
diffeomorphisms are isotopic if they are connected in the space Diff, with
C^{∞} 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]; 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.

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25 dec 2016