Notes

Here are some notes I’ve completed independently, drawing on various online resources. I hope they can make life a bit easier for others by:

• Offering detailed and thorough proofs of seemingly trivial or well-known results.

Homotopy Group of a Loop Space

• Through Loop-Suspension Adjunction, the following relationship for the homotopy group of a loop space $\Omega X$ is established: \(\pi_{k+1}(X) = \pi_k(\Omega X).\) This property implies that to determine the $k$-th homotopy group of $\Omega X$, it is sufficient to know the $(k+1)$-th homotopy group of $X$.
• Additionally, for an $n$-th loop space $\Omega^n X$, the homotopy group would be $\pi_{k+n}(X)$, derived by applying the Loop-Suspension Adjunction $n$ times.