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

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.

36 pages, 5 figures, substantial changes, the value of the cosmological constant in agreement with measurements, accepted in Physics of the Dark Universe

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