Homotopía

Homotopía is a fundamental concept in algebraic topology that describes a continuous deformation between two continuous maps between topological spaces. Informally, two maps are considered homotopic if one can be smoothly "stretched" or "shrunk" into the other without tearing or creating new holes.

More formally, if $f$ and $g$ are two continuous maps from a topological space $X$ to a topological space $Y$, then $f$ is homotopic to $g$, denoted by $f \simeq g$, if there exists a continuous map $H: X \times [0, 1] \to Y$ 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 variable $t \in [0, 1]$ can be thought of as a parameter representing time.

The relation of homotopy is an equivalence relation. This means it is reflexive ($f \simeq f$), symmetric ($f \simeq g$ implies $g \simeq f$), and transitive ($f \simeq g$ and $g \simeq h$ implies $f \simeq h$). This equivalence relation partitions the set of all continuous maps from $X$ to $Y$ into disjoint equivalence classes called homotopy classes.

The concept of homotopy is crucial for defining homotopy equivalence between topological spaces, which is a weaker notion of topological equivalence than homeomorphism. Two spaces are homotopy equivalent if there exist continuous maps between them that are "inverses" up to homotopy. This allows mathematicians to classify topological spaces based on their "shape" rather than their precise geometric structure. Homotopy theory is a major branch of mathematics that extensively studies these properties.