The moduli space of parallelizable 4-manifolds

TL;DR

This paper constructs a homotopy model for the space of smooth 4-manifolds, analyzing its discriminant hypersurface and extending invariants like Bauer-Furuta to parametrized families, introducing finite type invariants.

Contribution

It provides a homotopy-theoretic framework for the space of 4-manifolds and extends invariants to parametrized families, including the concept of finite type invariants.

Findings

Homotopy model for 4-manifold space constructed

Bauer-Furuta invariant extended to parametrized families

Introduction of finite type invariants with examples

Abstract

In this paper we construct the space of smooth 4-manifolds and find the homotopy model for the connected components of the complement to the discriminant. The discriminant of this space is a singular hypersurface and its generic points correspond to manifolds with isolated Morse singularities. These spaces can be considered as a natural base for the recent theories studying invariants for families. We show that the theory of Bauer and Furuta can be raised to parametrized families on our configurational space and their invariant is the step-function on chambers. We also introduce the definition of the invariant of finite type and give a simple example of an invariant of order one.