# 42. Loop spaces and reduced suspensions

In this lecture we define *H-groups*** **and *H-cogroups*, and prove that these are precisely the (co)group objects in the category $ \mathsf{hTop}_*$.

We then show that the *loop space*** **$ \Omega X$ of a pointed space $X$ is always an H-group, and that the *reduced suspension*** **$ \Sigma X$ of a pointed space $X$ is always an H-group.

Along the way we prove that $( \Omega, \Sigma)$ form an adjoint pair of functors on $ \mathsf{hTop}_*$.

On Problem Sheet Q you will prove that $S^n \cong \Sigma S^{n-1}$ for all n≥1. Thus the set $[S^n ,X]_*$ of pointed homotopy classes of maps $S^n \to X$ carries a group structure for $n \ge 1$. For $n=1$ this is the *fundamental group*** **that we studied in Lecture 4. For $n \ge 2$ these are the *higher homotopy groups*** **that we will study next lecture.

Comments and questions?