Infinite loop space

A space $ X$ is an infinite loop space if there is a sequence $ X_{0}=X,X_{1},\dots$ with homotopy equivalences $ X_{n} \rightarrow \Omega X_{n+1}$, i.e. $ X$ can be de-looped arbitrarily many times. By adjunction, we have maps $ \Sigma
X_{n} \rightarrow X_{n+1}$ so $ \{X_{n}\}$ form a spectrum. Conversely there is a functor from spectra to infinite loop spaces given by sending a spectrum $ \mathcal{E}$ to $ \lim \Omega^{n} E_{n}$.

Jeffrey Herschel Giansiracusa 2005-06-27