nLab
algebraic topology
Contents
Idea
Algebraic topology is generally the study of functors from nice categories of spaces to algebraic categories. This is in hindsight, in fact, category theory originally developed out of algebraic topology, where it was used first simply to describe what was going on and then to axiomatise Eilenberg-Steenrod cohomology theories.
- basic problems of algebraic topology, topology, differential topology
- homotopy theory, shape theory, nonabelian algebraic topology, rational homotopy theory
- homotopy lifting property, Hurewicz fibration, Hurewicz connection, Serre fibration
- homotopy extension property, Hurewicz cofibration, deformation retract
- suspension, loop space, mapping cylinder, mapping cone, mapping cocylinder
- cohomology, spectrum, Brown representability theorem
- fundamental group, fundamental groupoid
- homotopy group, Eckmann-Hilton duality, H-space, Whitehead product
- topological K-theory, complex cobordism, elliptic cohomology, tmf
- CW complex, CW approximation, simplicial complex, simplicial set
- model category, model structure on topological spaces, homotopy category
- fibration sequence, cofibration sequence
- Freudenthal suspension theorem, Whitehead theorem
References
A textbook with an emphasis on homotopy theory is in
- Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002) (pdf)
Lecture notes include
Brief indications of open questions and future directions (as of 2013) of algebraic topology and stable homotopy theory are in
Further online resources include
Revised on March 7, 2014 08:09:48
by
Urs Schreiber
(89.204.139.248)