The realization of a semisimplicial bundle map is a 𝑘-bundle map

Abstract

0. Outline of main results. Let p: E -B be an ss fiber bundle map [2] with group r and fiber F. We will prove that the geometric realization [5] IPI: lEt IBI has a fiber-bundle like structure. The easiest way to describe it is to call it a fiber bundle in the category of by which we mean the following. A Hausdorff space X is a k-space if it has the weak topology with respect to its compact subsets. A mapf of a k-space X into any space Y is continuous if and only if its restriction to each compact subset is continuous. Each CW complex is a k-space. Let (k denote the category of k-spaces and continuous maps. It is easy to check that the category Ck iS closed under finite products. If X, Y are k-spaces, then the category Xx kY is the space retopologized with the weak topology with respect to compact subsets. Note that if X, Y are CW complexes, then Xx k Y is always a CW complex, the one whose cells are the products of those of X by those of Y, and whenever the space Xx Y is a CW complex, then Xx k Y= X x Y. Using this definition of category product, we can define in the usual way a kgroup (a group in the category (k), and the operation in (k of the k-group r on the k-space F. 0.1. DEFINITION. A map p: X -Y in (k is a coordinate k-bundle map (=fiber bundle in (k) if there is a covering of Y by the interiors of closed neighborhoods {Uj}, coordinate functions Oj: Uj x F -p l(Uj), transition functions gij: Ui nl Uj F, etc., as in Steenrod [6], except that group, operation, product always are meant in the sense of (k It is easy to see from [5] and the definition of k-product, etc., that for any ss complexes X, Y, that I Xx YI = Il x k I YI; that the realization of an ss group is a k-group; and that if the ss group G operates on F, that I GI operates on IFt in the category k Our interest is in bundle structures which are closely related to the given ss structure. For example: