Space-time foam dense singularities and de Rham cohomology

TL;DR

This paper extends the abstract differential geometry framework to include dense and large singularities, enabling the construction of de Rham cohomology in highly singular space-time models inspired by general relativity.

Contribution

It significantly broadens the class of singularities in differential geometry, allowing for dense and large sets of singularities with a well-defined cohomology theory.

Findings

De Rham cohomology can be constructed on sets with dense singularities.

Singularities can have larger cardinality than nonsingular points.

The framework applies to space-time foam structures in general relativity.

Abstract

In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrary large positive Lebesgue measure. As also shown, one can construct in such a singular context a de Rham cohomology, as well as a short exponential sequence, both of which are fundamental in differential geometry. In this paper, these results are significantly strengthened, motivated by the so called space-time foam structures in general relativity, where singularities can be dense. In fact, this time one can deal with singularities on arbitrary sets, provided that their complementaries are dense, as well. In particular, the cardinal of the set of singularities can be larger than that of the nonsingular points.