Homotopieinvariante

Homotopieinvariante refers to a property of mathematical objects, particularly topological spaces, that remains unchanged under continuous deformations. In essence, if two spaces are of the same homotopy type, meaning one can be continuously deformed into the other, then any property that is a "homotopieinvariante" will be the same for both spaces. This concept is fundamental in algebraic topology, where it allows mathematicians to classify and distinguish between topological spaces that might look geometrically different but share deeper structural similarities.

Common examples of homotopieinvariante include the fundamental group, homology groups, and cohomology groups. These algebraic structures capture essential topological features, such as the presence of holes or connected components, and their properties are preserved even when a space is stretched, squeezed, or otherwise continuously deformed without tearing or gluing. For instance, a sphere and a cube are homotopicaly equivalent because one can be continuously deformed into the other; they both have trivial homology groups. Conversely, a torus (doughnut shape) and a sphere are not homotopicaly equivalent because the torus has a hole that cannot be eliminated by continuous deformation, and this difference is reflected in their differing homology groups. The study of homotopieinvariante provides a powerful lens through which to understand the underlying structure of topological spaces.