Configuration spaces in algebraic topology

Abstract

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the integral cohomology of the ordered---and the mod $p$ cohomology of the unordered---configuration spaces of $\mathbb{R}^n$, and the rational cohomology of the unordered configuration spaces of an arbitrary manifold of odd dimension. We also discuss models for mapping spaces in terms of labeled configuration spaces, and we show that these models split stably. Some classical results are given modern proofs premised on hypercover techniques, which we discuss in detail.