| Let M be the disjoint union of m copies of Rn (m ≥ 1, n ≥ 0), and let M0 denote the subspace consisting of the disjoint union of the closed unit balls. For a vector space V, we define AffEmb(M, V) to be the space of maps f : M → V which are affine on each component and which are embeddings when restricted to M 0. In this paper, we present a homotopy limit description of Sigma infinity AffEmb(M, V). Specifically, we begin by defining a collection of objects we call "complete affine partitions." Such an entity consists of a disjoint union of affine spaces, A = ⨿i Ai, along with an equivalence relation such that if two points lie in the same component of A, they are in the same equivalence class. Given a complete affine partition on A, we can consider maps f : A → M for which there is a pair of points in the same equivalence class but with distinct images under f; we call such maps "non-locally constant". We establish a category whose objects are pairs consisting of a complete affine partition and a non-locally constant map to M. From this category we consider the functor which takes such a pair to the suspension spectrum of the space of non-locally constant maps to V. In this paper we demonstrate that Sigmainfinity AffEmb(M, V) is the homotopy limit of this functor, over this category. To attain this, we make a digression to talk about categories whose objects and morphisms both form spaces, instead of just sets, as this is the case for our category.;This work extends the work of Arone, who has established preliminary results in the case when m = 1 (the case of linear injective maps, [2]) and the case when n = 0 (configuration spaces, [3]). Our method of proof is to reduce the category we construct to a category which is the join of the categories Arone uses. We also, therefore, describe the interaction of homotopy limits with joins of categories and functors. |
