Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

While algebraic K-theory has many applications and uses in modern algebra and arithmetic, its origins actually lie in geometric topology through Whithead's work on simple homotopy theory. We will review this in modern language, to elucidate the nature of simple homotopy types. This will also lead to parametrized versions of results of West and Chapman through the use of Efimov-K-Theory. The key is a K-theoretic model of assembly maps which is of independent nature and should have many more applications in the furture. This is joint work with A. Bartels and A. Efimov.