Partition complexes, Tits buildings and symmetric products (Q2766391)

Let \(\mathbf{n}\) denote the finite set \(\{1,2,\ldots ,n\}\). The collection of all nontrivial equivalence relations on \(\mathbf{n}\), partially ordered by refinement, has an associated space \(\mathbf{P}_n\) called the partition complex of \(\mathbf{n}\). On the other hand, assume that \(n=p^k\) (\(p\) a prime), next consider the associated Tits building \(T_k\) associated to the vector space \(\Delta=(\mathbb{F}_p)^k\). The authors show that there is a map \(T_k\to P_n\) equivariant with respect to the natural action of the affine group \(Aff_{k,p}= GL_{k}(\mathbb{F}_p)\ltimes \Delta\) and the symmetric group \(\Sigma_n\), respectively. The authors analyze this map in various contexts using homology approximations. Their results give connections between the homology of symmetric groups and that of affine groups, for example, they prove the following par Theorem 1.1. Suppose \(n>1\) is an integer. If \(n\) is not a power of \(p\), then \(\widetilde H^{\Sigma_n}_*(P_n^\lozenge; \mathbb{F}^\pm_p) \) vanishes. If \(n=p^k\), then the map \(\mathbf{T}_k\to\mathbf{P}_n\) induces an isomorphism NEWLINE\[NEWLINE\widetilde H^{Aff_{k,p}}_*\mathbf{T}_k^\lozenge;\mathbb{F}^\pm_p)\cong \widetilde H^{\Sigma_n}_*(\mathbf{P}_n^\lozenge;\mathbb{F}^\pm_p). NEWLINE\]NEWLINE Here \(\mathbb{F}_n^\pm\) stands for the sign representation of a given group on \(\mathbb{F}_p\) and \(X^\lozenge\) denotes the unreduced suspension of \(X\). The authors also find a relation between the partition complex and symmetric powers of spheres and explain why the homology of symmetric powers is related to the Steinberg idempotent. Furthermore, they provide a proof of a conjecture by Arone and Mahowald about the layers in the Goodwillie tower of the identity functor using these techniques.