On Topology of the Infinite-Dimensional Space of Fibrations

Abstract

This work serves as an opening and basis of an ongoing program investigating topological and geometric aspects of the moduli space of smooth fiberings on a manifold. The present paper focuses on the algebraic and differential topology of this space, and particularly addresses the following three quests in a top-down manner: the classification, for each class the path components, and for each component the homotopy type (as loosely analogous to the those three for studying the moduli of smooth structures: exotic manifolds, mapping class groups, and Smale-type conjectures). The last of the three is infinite-dimensional in nature, as the corresponding moduli space is shown to inherit the structure of a smooth Fr\'echet manifold from the diffeomorphism group through a (infinite-dimensional) principal bundle, with which we establish further connections with the Lie theory of gauge symmetries from one perspective, and with the geometric analysis of extrinsic flows from another. Concretely, we tackle the problem of finding the "homotopy core": a minimal deformation retract that encodes the topological structure of such a moduli space of fibrations; as our first examples, we gave explicit homotopy calculations for various low-dimensional cases which, combined with earlier known cases from others' work, complete the solution to this problem for dimensions up to three.