homotopy (idea)

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 S¹ to itself. If f: S¹ → S¹ 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 S¹ 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 S¹ 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 f₀, f₁ 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 f₀ to f₁) which satisfies H(x, 0) = f₀(x) and H(x, 1) = f₁(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 Rⁿ, is homotopy equivalent to a space consisting of a single point, but the invariance of domain theorem says that R^m and Rⁿ 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).