For a closed PL manifold , we consider the configuration space of ordered –tuples of distinct points in . We show that a suitable iterated suspension of is a homotopy invariant of . The number of suspensions we require depends on three parameters: the number of points , the dimension of and the connectivity of . Our proof uses a mixture of Poincaré embedding theory and fiberwise algebraic topology.
"On the homotopy invariance of configuration spaces." Algebr. Geom. Topol. 4 (2) 813 - 827, 2004. https://doi.org/10.2140/agt.2004.4.813