The space of retractions of a $2$-manifold

Abstract:

Let ${M^2}$ be a 2-manifold and let $\Lambda$ be the embedding of ${M^2}$ into its space of retractions which maps each point to the constant retraction to that point. Denote by $\mathcal {L}({M^2})$ the component containing the image of $\Lambda$. The embedding $\Lambda$, with range restricted to $\mathcal {L}({M^2})$, is shown to be a weak homotopy equivalence if ${M^2}$ is compact, or if ${M^2}$ is complete and the metric topology is used.