Introductory remark: This write-up is intentionally light on definitions; follow the hard-links for details.
A topological space is a space with a basic notion of shape, sufficient to define the notion of a continuous function. Two topological spaces can be considered equivalent if there is a homeomorphism between them, i.e. an invertible function which is continuous in both directions. A basic problem in topology is to determine whether two spaces are homeomorphic. If they are, then that can be proved by finding a homeomorphism. It is less obvious how to show that a pair of spaces is not homeomorphic. The easiest way to do it is usually to find some invariant that distinguishes between the spaces.
One simple property of topological spaces that is invariant under homeomorphisms is connectedness. Thus, if a pair of spaces has a different number of connected components then they cannot possibly be homeomorphic. But obviously this invariant is rather limited.
One basic purpose of algebraic topology is to define algebraic invariants which can be used to distinguish between topological spaces. This means that one associates an abstract algebraic object, like a group or a ring, to each topological space, and homeomorphic spaces should be associated to isomorphic objects. To be of any use, it should also not be too difficult to compute the invariants. Some of the most important invariants in algebraic topology of a space X are:
- The fundamental group π1X. This is the set of equivalence classes of closed loops in the space up to homotopy (continuous deformation), which forms a group under concatenation. It is trivial when the space is simply-connected, i.e. when all closed curves can be contracted to a point.
- Homology groups Hn(X). For each positive integer m, the degree m homology of a topological space is an abelian group. It can be vaguely thought of as counting the number of holes of dimension m in the space (in degree 0 it counts the number of connected components). For example, the homology of the n-dimensional sphere Sn has rank 1 in degree 0 and n, and rank 0 in all other degrees.
- Cohomology groups Hn(X). This is a dual concept to the homology groups; it's not really possible to explain the difference without going into detailed definitions. One important point is that it is possible to define a product of elements in the cohomology. The degree of the product is the sum of the degrees of the factors. Thus the sum of all the cohomology groups forms a graded ring.
- Higher homotopy groups πmX. The degree m homotopy group of a topological space is the set of homotopy classes of maps from Sn to the space. The degree 1 homotopy group is just the fundamental group, which behaves rather differently to the homotopy groups of higher degree.
The fundamental and (co)homology groups can be computed algorithmically given, say, a cell decomposition of a space. The higher homotopy groups are actually very difficult to compute, even for simple spaces like spheres.
The algebraic invariants listed above have nice `functorial´ properties in common: to a continuous map f between topological spaces X and Y there are natural associated maps (homomorphisms in fact) between the invariants of X and Y. Moreover, these homomorphisms are invariant under homotopies, i.e. deformations, of f. One consequence is that spaces that are homotopy equivalent have isomorphic invariants. For the purposes of deciding whether two topological spaces are homeomorphic this is a little inconvenient, since it means that homotopy equivalent spaces cannot be distinguished by the algebraic invariants. See cjeris's write-up on homotopy for some more discussion on this.
These algebraic invariants are defined for any topological space, including nasty singular ones. This is often convenient, but nevertheless it is most interesting to study the topology of `nice´ spaces. To me at least that means manifolds, i.e. topological spaces which locally look like flat space.
- Poincaré duality defines an isomorphism between the degree k homology and the degree n-k cohomology of a closed oriented manifold of dimension n.
- Morse theory tells a lot about the topology of a differentiable manifold, given any non-degenerate function on the manifold. Essentially, the homology of the manifold can be built up from information about the critical points of the function.
- For differentiable manifolds one can define de Rham cohomology, a cohomology theory in terms of differential forms. This is isomorphic to the usual cohomology (with real coefficients) of the manifold. This way statements about the differential geometry of the manifold can be translated into topological properties, and vice versa.
- For a Riemannian manifold, i.e. a differentiable manifold with a way of measuring lengths, Hodge theory shows that the de Rham cohomology is isomorphic to the spaces of harmonic forms, which gives a link between analysis and topology.
- The Poincaré conjecture could be described as stating that any closed 3-dimensional manifold that has the same algebraic invariants as the 3-sphere must in fact be homeomorphic to the 3-sphere. In other words, if one restricts attention to manifolds then it is sometimes true that homotopy equivalence implies homeomorphism. This is a very hard problem, in fact one of the Millennium Prize problems. Interestingly, Perelman's solution to this topological problem takes a large detour into geometry.
Some consider that the `nice´ spaces one should be interested in are varieties, the basic objects of study in algebraic geometry. They are wrong, of course, but some of their ideas are still useful. Thinking about the topology of varieties over arbitrary fields is more pain than I can handle, but if one works over the complex numbers then they define reasonable spaces (when non-singular they are complex manifolds).
- The Lefschetz hyperplane theorem in its simplest form states that intersecting an n-dimensional projective variety with a hyperplane does not change its (co)homology or homotopy groups up to degree n-1. I like this theorem because the statement is simple enough that even I can apply it when I need to figure out the topology of a projective variety.
- The rigid structure of an algebraic variety allows the definition of many different sheaves. For some of these the associated sheaf cohomology is related to the usual cohomology.
- Dolbeault cohomology is a version of de Rham cohomolgy for complex manifolds. It is defined in terms of differential forms, and is isomorphic to the cohomology of the sheaf of holomorphic forms.
- If a complex manifold has a Kähler metric, i.e. a Riemannian metric compatible with the complex structure, then Hodge theory can be used to identify Dolbeault cohomology spaces with subspaces of the usual cohomology. This gives rise to the Hodge decomposition of the cohomology Kähler manifold. The important point is that this is independent of the choice of metric, so it is a property of the complex manifold itself.
- On a Kähler manifold the Hodge decomposition yields a necessary condition for which elements of the rational cohomology of the manifold can be represented by sub-varieties. The Hodge conjecture states that for non-singular projective varieties (which are always Kähler manifolds) this condition it also sufficient. In slogan form: a certain part of the algebraic topology of the variety can be described completely in terms of its algebraic geometry. This is another of the Millennium Prize problems.