On connectedness properties of complement of closed hausdorff weakly infinite-dimensional subset in the moduli space of all complete riemannian metrics on the plane

Abstract

In Riemannian geometry, introducing geometric concepts to smooth manifolds is done via selecting an appropriate Riemannian metric. We define 𝑅≥0 (𝑅 2 ) to be the space of all complete Riemannian metrics of non-negative curvature on the plane. The Lie group 𝐷𝑖𝑓𝑓(𝑅² ) of all self diffeomorphisms onto 𝑅² acts on 𝑅≥0 (𝑅² ) by pulling back metrics. Denote the moduli space of all complete Riemannian metrics of non-negative curvature on the plane by 𝑀≥0 (𝑅² ), it is the quotient space of 𝑅≥0 (𝑅² ) by the 𝐷𝑖𝑓𝑓(𝑅² ) action via pullback. The moduli space 𝑀≥0 (𝑅² ) is not a manifold since different Riemannian metrics may have isometry groups of different dimensions. A topological space 𝑋 is said to be weakly infinite-dimensional if for every family {(𝐴𝑖 , 𝐵𝑖 ):𝑖 ∈ 𝑁} of pairs of disjoint closed subsets of 𝑋, there exist separators 𝐷𝑖 between 𝐴𝑖 and 𝐵𝑖 such that ⋂𝑖=1 ∞ 𝐷𝑖 = ∅. The connectedness properties of the space 𝑅≥0 (𝑅² ) and 𝑀≥0 (𝑅² ) were first studied by Belegradek and Hu, and they proved that the complement of every finitedimensional subset of the space 𝑅≥0 (𝑅² ) is continuum-connected. It was later proved that the complement of every closed, finite-dimensional subset of 𝑅≥0 (𝑅² ) is path-connected and that the complement of a subset of 𝑀≥0 (𝑅² ) is path-connected if the subset is countable, or it is closed, metrisable and finite-dimensional. The results for 𝑅≥0 (𝑅² ) were generalised to show that complement of every closed, weakly infinite-dimensional subset of 𝑅≥0 (𝑅² ) is pathconnected. Further, a partial generalisation on 𝑀≥0 (𝑅² ) was obtained to prove that the complement of a closed Hausdorff space with Haver’s property 𝐶 of 𝑀≥0 (𝑅² ) is pathconnected. In this research, we prove that the complement of every closed Hausdorff weakly infinite-dimensional subset of 𝑀≥0 (𝑅² ) is path-connected, with an argument using a dimension theoretic argument on the dimensionality of a paracompact preimage of a fully closed map onto a weakly infinite-dimensional space. With this result, we conclude the series of theorems of connectedness properties of 𝑅≥0 (𝑅² ) and 𝑀≥0 (𝑅² ).