Shape Theory

TL;DR

Shape theory, founded by K. Borsuk, is a significant area in geometric topology focusing on spectral homotopy theory, with foundational concepts and key results summarized in this survey.

Contribution

This paper provides an overview of the basic concepts and important results in shape theory, highlighting its role in geometric topology.

Findings

Summarizes fundamental concepts of shape theory

Highlights key results in spectral homotopy theory

Provides extensive bibliography for further study

Abstract

Shape theory was founded by K.~Borsuk 50 years ago. In essence, this is spectral homotopy theory; it occupies an important place in geometric topology. The article presents the basic concepts and the most important, in our opinion, results of shape theory. Unfortunately, many other interesting problems and results related to this theory could not be covered because of space limitations. The article contains an extensive bibliography for those who wants to gain a more detailed and systematic insight into the issues considered in the survey.