Cobe Satellite Measurement, Hyperspheres, Superstrings and the Dimension of Spacetime

The first part of the paper attempts to establish connections between hypersphere backing in infinite dimensions, the expectation value of dim E (∞) spacetime and the COBE measurement of the microwave background radiation. One of the main results reported here is that the mean sphere in S (∞) spans a four dimensional manifold and is thus equal to the expectation value of the topological dimension of E (∞) . In the second part we introduce within a general theory, a probabolistic justification for a compactification which reduces an infinite dimensional spacetime E (∞) (n=∞) to a four dimensional one (DT=n=4) . The effective Hausdorff dimension of this space is given by fn5dimH E (∞) =dH =4+φ3 where φ3 =1/ [4+φ3 ] is a PV number and is the Golden Mean. The derivation makes use of various results from knot theory, four manifolds, noncommutative geometry, quasi periodic tiling and Fredholm operators. In addition some relevant analogies between E (∞) , statistical mechanics and Jones polynomials are drawn. This allows a better insight into the nature of the proposed compactification, the associated E (∞) space and the Pisot-Vijayvaraghavan number 1/φ3=4.236067977 representing its dimension. This dimension is in turn shown to be capable of a natural interpretation in terms of Jones knot invariant and the signature of four manifolds. This brings the work near to the context of Witten and Donaldson topological quantum field theory.