Shape theory (mathematics)

Shape theory is a branch of topology that studies the global geometric properties of compact metric spaces, particularly those properties that are preserved under "shape equivalences." It provides a coarser classification of spaces than homotopy theory, focusing on the overall shape rather than the detailed local structure.

Unlike homotopy theory, which works well for spaces with good local properties (like ANRs - Absolute Neighborhood Retracts), shape theory is designed to handle spaces with potentially bad local properties, such as fractals and pathological continua. These spaces often have trivial homotopy groups, making homotopy theory inadequate for distinguishing them.

The fundamental idea of shape theory involves approximating a given space X by a sequence of ANRs. A "shape" is then defined as an equivalence class of systems of maps from these ANRs into X. This approach allows shape theory to capture the overall connectedness and "shape" of a space, even when it has complicated local behavior that obstructs standard homotopy methods.

Shape equivalences are weaker than homotopy equivalences; that is, spaces that are homotopy equivalent are also shape equivalent, but the converse is not necessarily true. This makes shape theory a useful tool for classifying spaces that are distinguishable by their global shape but not by their homotopy type.

Key concepts in shape theory include:

• Shape category: A category where objects are topological spaces and morphisms are shape morphisms.
• Shape functor: A functor that maps a topological space to its shape (an object in the shape category).
• FANR (Fundamental Absolute Neighborhood Retract): A space that is shape equivalent to an ANR.

Shape theory has connections to other areas of mathematics, including continuum theory, geometric topology, and dynamical systems. It provides tools for studying spaces with complex topological structure and has applications in areas such as image analysis and computer graphics.