Spaces of arcs and configuration spaces of manifolds (Q1344454)

The relation between the homotopy type of a smooth manifold \(M\) and the homotopy type of the space of embeddings of some 0- and 1-dimensional spaces into \(M\) (configuration space and space of arcs respectively) is examined. It is proved that the homotopy type of the loop space of the configuration space and the space of arcs of \(M\) depend only on the homotopy type of \((M, \partial M)\). This result gives further strong evidence for a long-standing conjecture that the homotopy type of the configuration space of \(M\) depends only on the homotopy type of \((M, \partial M)\).