The subject of

algebraic topology is concerned with finding discrete, effectively

computable invariants which distinguish one

topological space or

continuous map from another. A simple example is the

degree of a

map from the circle S

^{1} to itself. If f: S

^{1} → S

^{1} is a

continuous map, there is a well-defined integer deg(f) which, intuitively speaking, counts the number of times the loop f winds around the target circle. (If f is

smooth, deg(f) is easy to compute: it's the

integral of the pulled-back angle

form f

^{∗}dθ, divided by 2π.) But there are only

countably many integers, and the space of

continuous maps from S

^{1} to itself is very large (

uncountable at least), so

degree is certainly not a complete description of such a map. It turns out that the

degree classifies

maps from S

^{1} to itself up to homotopy: two such

maps have the same

degree if and only if one of them is continuously deformable into the other. This situation reproduces itself throughout

algebraic topology. There is essentially no hope of finding

computable invariants of spaces and maps if we want to detect the

*exact* space or map with the

invariant; there are just too many

continuous maps to put in any kind of sensible algebraic structure. But if we agree to consider two maps the same if they are homotopic, that is, one of them is continuously deformable into the other, then there is a very rich theory which can distinguish many different

spaces via algebraic

invariants called

homology groups and

homotopy groups.

Formally, we say that continuous maps f_{0}, f_{1} from a space X to a space Y are *homotopic* if there is a continuous map H: X × [0,1] → Y (called a *homotopy* from f_{0} to f_{1}) which satisfies H(x, 0) = f_{0}(x) and H(x, 1) = f_{1}(x) for every x ∈ X. If X and Y are pointed spaces, that is we have fixed base points ∗_{X} ∈ X and ∗_{Y} ∈ Y, then we suppose also that H(∗_{X}, t) = ∗_{Y} for every t ∈ [0,1] (that is, H is a homotopy *relative to ∗*_{X}). (The concept of a pointed space is a technical kludge which turns out to be almost mandatory in algebraic topology.) In general if A ⊂ X, a homotopy *relative to A* is one for which H(a, t) = a for every a ∈ A and t ∈ [0,1]. The relation of homotopy is conventionally denoted by a symbol which is not available in HTML; it looks like an equals sign where the top bar is a tilde, halfway between ∼ and ≅.

Two spaces X and Y are called homotopy equivalent if there are maps f: X → Y and g: Y → X so that gf is homotopic to the identity map of X, and fg is homotopic to the identity map of Y. The relation of homotopy equivalence is *much* coarser than that of homeomorphism: for instance, any contractible space, such as Euclidean space **R**^{n}, is homotopy equivalent to a space consisting of a single point, but the invariance of domain theorem says that **R**^{m} and **R**^{n} are homeomorphic only if m = n.
(Invariance of domain is itself not an easy theorem.) Nevertheless, to get some of the most powerful and interesting results of algebraic topology, we are forced to consider spaces only up to homotopy equivalence.

For more algebraic topology on E2 see homology groups. To learn more, try the second edition of *Topology: a first course* by James Munkres (if you don't already know much point-set topology), or one of my favorites, *Topology and geometry* by Glen Bredon (if you do).