nLab
algebraic topology

Idea

Algebraic topology refers to the application of methods of algebra to problems in topology. More specifically, the method of algebraic topology is to assign homeomorphism/homotopy-invariants to topological spaces, or more systematically, to the construction and applications of functors from some category of topological objects (e.g. Hausdorff spaces, topological fibre bundles) to some algebraic category (e.g. abelian groups, modules over the Steenrod algebra). Landing in an algebraic category aids to the computability, but typically loses some information (say getting from a topological spaces with a continuum or more points to rather discrete algebraic structures). The basic aim is to attack a classification problem (for spaces or maps) or existence of maps (typically the problems for finding liftings, sections, extensions and retractions, cf. basic problems of algebraic topology) and, more rarely, the uniqueness problem for maps. With the development of the subject, however the invariants and the objects of algebraic topology are not only used to attack these problems but also to characterize the conditions and to have the language for various constructions (say vanishing conditions, conditions on characteristic class, dual classes, products).

The idea of functorial invariants

The basic idea of the functorial method for the problem of existence of morphisms is the following: If $F:A\to B$ is a functor (we present here a general statement, but in the above context $A$ is a category of topological objects and $B$ some category of algebraic objects) and $d:D\to A$ a diagram in $A$ then $F\circ d$ is a diagram in $B$. If one can fill certain additional arrow $f$ in the diagram $d$ making the extended diagram commutative, then $F(f)$ is a morphism between the corresponding vertices in $B$ extending $F\circ d$ to a commutative diagram. Thus if we prove that there is no morphism extending $F\circ d$ then there was no morphism extending $d$ in the first place. Therefore, the functorial method is very suitable to prove negative existence for morphisms. Sometimes, however, there is a theorem showing that some set of invariants completely characterizes a problem hence being able to show positive existence or uniqueness for maps or spaces. For the uniqueness for morphisms, it is enough to show that $F$ is faithful and that there is at most one solution for the existence problem in the target category. Faithful functors in this context are rare, but it is sufficient for $F$ to be faithful on some subcategory $A_p$ of $A$ containing at least all morphisms which are the possible candidates for the solution of the particular existence problem for morphisms.

Overview of methods

The archetypical example is the classification of surfaces via their Euler characteristic. But as this example already shows, algebraic topology tends to be less about topological spaces themselves as rather about the homotopy types which they present. Therefore the topological invariants in question are typically homotopy invariants of spaces with some exceptions, like the shape invariants for spaces with bad local behaviour.

Hence modern algebraic topology is to a large extent the application of algebraic methods to homotopy theory.

A general and powerful such method is the assignment of homology and cohomology groups to topological spaces, such that these abelian groups depend only on the homotopy type. The simplest such are ordinary homology and ordinary cohomology groups, given by singular simplicial complexes. This way algebraic topology makes use of tools of homological algebra.

The axiomatization of the properties of such cohomology group assignments is what led to the formulation of the trinity of concepts of category, functor and natural transformations, and algebraic topology has come to make intensive use of category theory.

In particular this leads to the formulation of generalized (Eilenberg-Steenrod) cohomology theories which detect more information about classes of homotopy types. By the Brown representability theorem such are represented by spectra (generalizing chain complexes), hence stable homotopy types, and this way algebraic topology comes to use and be about stable homotopy theory.

Still finer invariants of homotopy types are detected by further refinements of these “algebraic” structures, for instance to multiplicative cohomology theories, to equivariant homotopy theory/equivariant stable homotopy theory and so forth. The construction and analysis of these requires the intimate combination of algebra and homotopy theory to higher category theory and higher algebra, notably embodied in the universal higher algebra of operads.

The central tool for breaking down all this higher algebraic data into computable pieces are spectral sequences, which are maybe the main heavy-lifting workhorses of algebraic topology.

Related entries

• 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

Textbooks include

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

• Peter May, A concise course in algebraic topology

• Peter May, Kate Ponto, More concise algebraic topology

• Allen Hatcher, Algebraic Topology

Lecture notes include

• Michael Hopkins (notes by Akhil Mathew), algebraic topology – Lectures (pdf)

• Friedhelm Waldhausen, Algebraische Topologie I (pdf) , II (pdf), III (pdf) (web)

• Davis, Lecture notes in algebraic topology (pdf)

A textbook with an emphasis on homotopy theory is in

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

A comprehensive survey of various subjects in algebraic topology is in

• Ioan Mackenzie James, Handbook of Algebraic Topology 1995

Further online resources include

• a thread on this at MathOverflow.

Brief indications of open questions and future directions (as of 2013) of algebraic topology and stable homotopy theory are in

• Tyler Lawson, The future, Talbot lectures 2013 (pdf)