55: Algebraic topology
Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants reflect some of the topological structure of the spaces. The use of these algebraic tools calls attention to some types of topological spaces which are well modeled by the algebra; fiber bundles and related spaces are included here, while complexes (CW-, simplicial-, ...) are treated in section 57. Finally, the use of the algebraic tools also calls attention to the aspects of a topological space which are well modeled by the algebra; this gives rise to homotopy theory.
The algebraic tools used in topology include various (co)homology theories, homotopy groups, and groups of maps. These in turn have necessitated the development of more complex algebraic tools such as derived functors and spectral sequences; the machinery (mostly derived from homological algebra) is powerful if rather daunting.
Arguably, the value of algebraic topology is measured by the extent to which it answers questions which arise more geometrically. These classical topics were, before the development of the full algebraic tools, often treated as questions involving numbers. For example, the dimension of a real vector space is a topological invariant (R^n is not homeomorphic to R^m if n > m) but this was not easy to prove for m > 1. Likewise, many questions about maps between n-dimensional spaces could be resolved with an appeal to the degree of the map (an extension of the winding number from complex analysis, due to Brouwer) -- the Fundamental Theorem of Algebra and the Hairy Ball Theorem are two well-known examples. Other questions of a geometric nature which can be addressed with an appeal to the algebra include fixed-point theorems (for example Lefschetz's theorem guarantees a fixed point if a certain calculated number is nonzero; Smith theory studies the regularity of fixed points under self-homeomorphisms of finite order).
Of the many algebraic constructs used in this discipline we might make a distinction between those which are somewhat more flexible but weaker, and those which are stronger but more complicated (e.g. they include non-abelian groups). In this scheme, the primary tools of the first sort are the (co)homology theories. These are functors assigning to each space (better: to each space-subspace pair) an abelian group, in a "natural" way. There are many useful examples of homology theories. Some differ from each other mostly in the means of definition, so that they give identical calculations for "nice" spaces (e.g. Cech and singular homologies). Some differ mostly in the sort of group constructed (e.g. H_n(X,A) for different choices of group A will always be zero if X is a point, for positive n; but choosing A to be the real line makes H_n(X,A) a vector space; choosing other A will make H_n(X,A) a p-group for all X; and so on). One has a similar "natural" behaviour even in cases where the "homology" of a point is not zero; these are the "extraordinary" homology theories, such as K-theory.
In all cases, the "naturality" of the construction implies that a map between spaces induces a map between the groups. Thus one can show that no maps of some sort can exist between two spaces (e.g. homeomorphisms) since no corresponding group homomorphisms can exists. That is, the groups and homomorphisms offer an algebraic "obstruction" to the existence of maps. Classic applications include the nonexistence of retractions of disks to their boundary and, as a consequence, the Brouwer Fixed-Point Theorem. (Obstruction theory is, more generally, the creation of algebraic invariants whose vanishing is necessary for the existence of certain topological maps. For example a function defined on a subspace Y of a space X defines an element of a homology group; that element is zero iff the function may be extended to all of X.)
Homology theories are the natural settings for intersection theory. Essentially a topic from algebraic geometry, this asks for a method of determining the greatest extent to which two subspaces of the space can avoid each other. Cohomology theories are a slight change from homology theories in that the directions of some homomorphisms are reversed; they're roughly the dual groups of the homology groups. These are the setting of product structures: collections of cohomology groups of a space form a cohomology ring, which of course has more structure than its underlying additive group. (The "cup" product in ordinary singular homology is dual to the "cap" product, or intersection pairing, in homology theory, linking these two themes.)
Increasingly strong geometric conclusions can be drawn from increasingly sensitive -- and complicated -- algebraic constructions attached to spaces. For example, the singular cohomology rings can be further given the structure of an algebra over a key, and complicated, ring known as the Steenrod algebra. (This is the collection of all possible natural homomorphisms between one homology functor and another -- that is, the set of "cohomology operations".) Other algebraic tools can be even more sensitive; for example, the secondary cohomology operations (defined on the kernel of the previous operations) can be used to prove certain spheres lack independent vector fields, a purely topological result which in turn implies that there are no real division algebras of degree greater than 8!
Homology groups are particularly well suited to computation via some inductive procedure: if a space is somehow pieced together from simpler spaces (as unions, say, or fibrations) then the homology theories of the large space reflect those of the smaller spaces, together with some algebraic information which indicates the nature of the piecing-together. The tools for combining these data are spectral sequences. While the general theory of their development is more properly a topic in homological algebra, certain spectral sequences are very commonly used in algebraic topology. (Precise attribution of credit is a bit difficult but Serre is usually associated with the singular homology spectral sequence in ordinary homology theories.)
Apart from homology groups and their kin, the principal algebraic tool used in topology is the set of homotopy groups of a space, and related concepts; in particular this includes the fundamental group (pi_1(X)) of a space. Since the fundamental group of a space need not be abelian it is clear that the study of these functors may be more difficult than the homology theories; on the other hand, they are more sensitive and powerful.
To give an indication of the difficulty, it is not known how to present in a straightforward way the set of homotopy groups of the 2-dimensional sphere! Indeed, spheres are in some sense the building blocks of topology, from the perspective of homotopy theory, and so their homotopy groups are particularly significant. (Quite a bit is known about specific families of these groups.)
Several geometric constructs are particularly well matched with homotopy groups. For example, one can relate the groups of a space and its suspension (suspension is a sort of dimension-raising tool). Also, the homotopy groups, like the homology groups, are unchanged by homotopy equivalences -- maps between spaces which include homeomorphisms but are far more general (they include maps which may be "deformed" to a homeomorphism). Thus one is led to a separate discipline, Homotopy Theory, which brings these features to the forefront.
Homotopy theory focuses on the most intrinsically invariable features of a space -- for example, all Euclidean spaces are in homotopy theory considered the same, since all are contractible; the circle is decidedly distinct, since it has a hole. If one goes further and blurs the distinction between spheres of different dimensions (S^n is the suspension of S^(n-1)) we have what is known as Stable Homotopy Theory. Clearly in this area we are distinguishing spaces only when they are quite fundamentally distinct. This permits a detailed investigation of just one or two features of a space. The process is analogous to localization and completion in a ring, and indeed, one may study the localization and completion of a space!
As indicated in the paragraphs above, the use of algebraic tools draws attention to certain kinds of spaces, and certain aspects of spaces, which are most easily studied with these tools. We have mentioned the topics of fibrations and fiber bundles -- these are spaces which are locally products of two other spaces, but not necessarily globally (as the Möbius band is a fibration of a circle by the real line). If the fiber space is discrete, a fibration is nothing but a covering space (e.g. a helix is a covering space of a circle). There is a very pretty Galois-like correspondence comparing covering spaces of a space and subgroups of its fundamental group. If the fiber space is a vector space, we have the theory of vector bundles as used in differential geometry. These are "classified" either algebraically (with "characteristic classes" -- elements of homology groups) or geometrically (via maps to a a "classifying space"). Other kinds of spaces arising naturally in this area include loop spaces (the space of loops in a given space); the loop-space functor is nearly an inverse to the suspension functor. Simplicial complexes and CW-complexes are discussed in section 57; these are spaces on which the usual homology theories are particularly well-behaved.
Many themes in algebraic topology also appear with the adjective "equivariant". This implies that there is a group acting on the space(s) and the maps -- geometric and algebraic -- among the objects are to commute with the action of the group. This is a natural extension of ordinary algebraic topology: ordinary functors on a space are essentially equivariant functors on its universal cover, the group acting on them being the fundamental group. While many results of ordinary algebraic topology carry over without real change, there are topics of interest which amount to a kind of representation theory for groups; tools from group theory can be heavily used.
See the article on Topology at St Andrews.
Thorough, excellent coverage is provided by Dieudonné, Jean, "A history of algebraic and differential topology. 1900--1960", Birkhäuser Boston, Inc., Boston, MA, 1989, 648 pp. ISBN 0-8176-3388-X MR90g:01029. He has a similar (shorter!) survey done a little later (see MR95k:55001)
Some later developments are discussed in Adams, J. F., "Algebraic topology in the last decade", in Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), pp. 1--22. Amer. Math. Soc., Providence, R. I., 1971. MR47 #5858
A recent preprint by Peter May covers Stable algebraic topology 1945-1966
There are two other topology pages; applications of the algebraic invariants of topology are often found in these other areas:
General topology focuses on the underlying spaces and is often concerned with fairly analytical issues (e.g. convergence of sequences) rather than the use of algebraic tools.
Manifolds and cell complexes focuses on some of the more geometric aspects of topology: actions of groups on spaces, knot theory, low-dimensional topology, fibrations, and so on. Roughly: this is the part of the topology most amenable to nice pictures. This includes differential topology -- what happens when we add differential or other structures? -- and the study of those spaces for which homology theories give the strongest results -- simplicial complexes and CW-complexes.
Simplicial complexes are essentially polyhedra, although the latter term is generally used when considering geometric aspects of the spaces.
The tools of algebraic topology, when developed in isolation or for applications to other fields such as ring theory, give rise to homological algebra and category theory; this is the proper framework for comparing different algebraic tools.
The field of K-theory has two aspects, algebraic and topological, which are linked at a very deep level (via classifying spaces for groups associated to rings). The topological end of K-theory is really the topic of vector bundles over spaces.
One often hears of "homotopy methods" in various fields of mathematics (e.g. numerical analysis); these typically use only the most elementary aspects of homotopy (the idea of a continuous deformation) but for example the topic of monodromy groups in complex variables or differential equations is essentially an illustration of the fundamental group.
Homological methods are used throughout algebra. While primarily a direct application of homological algebra, these are indirectly related to topology (e.g. cohomology of groups is essentially the same as cohomology of classifying spaces). Homological methods used in analysis more typically call for homology with coefficients in a sheaf of functions. Topics such as DeRham cohomology link the topological features with the analytic, as in global analysis.
Browse all (old) classifications for this area at the AMS.
An excellent description of the state of the art: "Handbook of algebraic topology", edited by I. M. James. North-Holland, Amsterdam, 1995. 1324 pp. ISBN 0-444-81779-4 (a set of survey articles by leading topologists.)
A good collection of survey articles: "Algebraic topology and its applications", edited by G. E. Carlsson, R. L. Cohen, W. C. Hsiang and J. D. S. Jones. Mathematical Sciences Research Institute Publications, 27. Springer-Verlag, New York, 1994. 267 pp. ISBN 0-387-94098-7 MR95b:55001
At the undergraduate level it is difficult to find texts which cover more of algebraic topology than the fundamental group, perhaps. There are several good, well-known graduate-level texts, including
Readers favoring analysis will surely want to read Jean Dieudonné's "Elements d'analyse. Tome IX. Chapitre XXIV", Cahiers Scientifiques XL11. Gauthier-Villars, Paris, 1982. 380 pp. ISBN 2-04-011499-8, MR84a:57021
There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period.
Among the texts which will be listed on index pages for the subareas, if they are created: 55R:Husemoller, Dale, "Fiber bundles", (3/e GTM 20) Springer-Verlag, New York, 1994. 353 pp. ISBN 0-387-94087-1, MR94k:55001
Online text(s) [Allen Hatcher]
There is a mailing list for algebraic topology. (I believe it is mostly an update-notification service for Clarence Wilkerson's Hopf Topology Archive.