How to obtain a cosmological constant from small exotic R^4

TL;DR

This paper proposes a novel method to derive the cosmological constant as a topological invariant using exotic smooth structures on R^4 and techniques from low-dimensional differential topology, linking geometry and topology.

Contribution

It introduces a way to compute the cosmological constant as a topological invariant associated with exotic R^4 structures embedded in standard R^4, utilizing complex surface constructions.

Findings

Cosmological constant can be expressed as a topological invariant.

Exotic R^4 structures admit hyperbolic geometry simplifying calculations.

The smallness of the cosmological constant is explained through topological invariants.

Abstract

In this paper we determine the cosmological constant as a topological invariant by applying certain techniques from low dimensional differential topology. We work with a small exotic $R^{4}$ which is embedded into the standard $\mathbb{R}^{4}$. Any exotic $R^4$ is a Riemannian smooth manifold with necessary non-vanishing curvature tensor. To determine the invariant part of such curvature we deal with a canonical construction of $R^4$ where it appears as a part of the complex surface $K3\#\overline{CP(2)}$. Such $R^{4}$'s admit hyperbolic geometry. This fact simplifies significantly the calculations and enforces the rigidity of the expressions. In particular, we explain the smallness of the cosmological constant with a value consisting of a combination of (natural) topological invariant. Finally, the cosmological constant appears to be a topologically supported quantity.