On Topology of the Infinite-Dimensional Space of Fibrations

TL;DR

This paper explores the topological and geometric structure of the infinite-dimensional moduli space of smooth fibrations, focusing on classification, path components, and homotopy types, with explicit calculations for low-dimensional cases.

Contribution

It introduces a framework for understanding the homotopy core of the moduli space of fibrations, including explicit calculations in low dimensions and connections to gauge theory and geometric analysis.

Findings

The moduli space inherits a smooth Fréchet manifold structure.

Explicit homotopy calculations are provided for dimensions up to three.

The homotopy core encodes the topological structure of the space.

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.