## Detecting Mapping Spaces and Spaces with Complexity One

##### Abstract

In this thesis we will prove two main results regarding a space A following from information about mapping spaces out of a space A. We show if A is a finite CW-complex such that algebraic theories detect mapping spaces out of A, then A has the homology type of a wedge of spheres of the same dimension. Furthermore, if A is simply connected then A has the homotopy type of a wedge of spheres.The A-cellular approximation of a space X is the space that contains the most information we can possibly recover from X given the mapping space from A to X. The A-complexity of a space X is an ordinal number that quantifies how difficult it is to build an A-cellular approximation of X. We show that if A is a space such that mapping spaces out of A can be described by the algebraic theory associated to A, then the A-complexity of any space X is at most 1. This holds in particular for spheres and spheres localized at a set of primes.