Remarks on finite subset spaces
Sadok Kallel, Denis Sjerve · Homology Homotopy and Applications · 2009
This paper expands on and refines some known and less well-known results about the finite subset spaces of a simplicial complex X including their connectivity and manifold structure.It also discusses the inclusion of the singletons into the threefold subset space and shows that this subspace is weakly contractible but generally non-contractible unless X is a cogroup.Some homological calculations are provided. Statement of resultsLet X be a topological space (always assumed to be path-connected), and k a positive integer.It has become increasingly useful in recent years to study the space 3,9,15,19,23].This space is topologized as the identification space obtained from X n by identifying two n-tuples if and only if the sets of their coordinates coincide [4].The functors Sub n (-) are homotopy functors in the sense that if X Y , then Sub n (X) Sub n (Y ).If k n, then Sub k X naturally embeds in Sub n X.We write j n : X ↩→ Sub n X for the inclusion given by j n (x) = {x}.This paper takes advantage of the close relationship between finite subset spaces and symmetric products to deduce a number of useful results about them.As a starting point, we discuss cell structures on finite subset spaces.We observe in Section 3 that if X is a finite d-dimensional simplicial complex, then Sub n X is an nd-dimensional CW-complex and of which Sub k X for k n is a subcomplex (Proposition 3.1).Furthermore, Sub X := n 1 Sub n X has the structure of an abelian CWmonoid (without unit) whenever X is a simplicial complex.In Section 4 we address a connectivity conjecture stated in [25].We recall that a space X is r-connected if π i (X) = 0 for i r.A contractible space is r-connected for all positive r.In [25] Tuffley proves that Sub n X is n -2-connected and conjectures that it is n + r -2-connected if X is r-connected.We are able to confirm his conjecture for the three-fold subset spaces.In fact we show Research