Algebraic K-theory of spaces

Summary

The algebraic K-theory of a space $X$, written $A(X)$, is a spectrum defined by Waldhausen from spaces built from $X$ by attaching finitely many cells. It is a version of algebraic K-theory in which rings are replaced by spaces. It links K-theory to the geometry of manifolds: through it, the spaces of h-cobordisms and of diffeomorphisms of high-dimensional manifolds are related to homotopy theory. For a point, $A(\ast)$ is the K-theory of the sphere spectrum.