Homotopien

Homotopien are a fundamental concept in algebraic topology used to study topological spaces. Intuitively, two continuous maps between topological spaces are considered homotopic if one can be continuously deformed into the other. More formally, two continuous maps f and g from a topological space X to a topological space Y are homotopic, denoted f ~ g, if there exists a continuous map H: X × [0,1] → Y such that for all x in X, H(x, 0) = f(x) and H(x, 1) = g(x). The map H is called a homotopy between f and g. The parameter t in [0,1] can be thought of as time, and H(x, t) represents the position of the point x at time t during the deformation.

Homotopy is an equivalence relation. This means it is reflexive (every map is homotopic to itself), symmetric (if f is homotopic to g, then g is homotopic to f), and transitive (if f is homotopic to g and g is homotopic to h, then f is homotopic to h). This equivalence relation partitions the set of all continuous maps between two spaces into equivalence classes called homotopy classes.

The concept of homotopy is crucial for defining homotopy groups, which are algebraic invariants of topological spaces. For example, the fundamental group, a key tool in algebraic topology, is defined using loops and their homotopies. Homotopy theory provides powerful methods for distinguishing between topological spaces that may have the same homology groups but are topologically distinct. It also plays a significant role in fields such as differential geometry and theoretical physics.