This repository contains a draft article which describes various connections and applications of Dold-Thom's theorem and optimal transportation. The article develops the following themes:
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A reformulation of DT where the homology groups of a space X are represented in the pointed homotopy groups of A_0(X), the space of signed distributions on
$X$ which have zero total sum. This space is like the net neutral electrical charge distributions on$X$ . The space A_0(X) is canonically basepointed at the zero element, or vacuum state φ. We develop a relative version of DT such that the long exact sequence in relative homology precisely corresponds to the long exact sequence of homotopy groups. This has not appeared in the literature before. -
We develop optimal transportation as a tool for generating geometric sweepouts of manifolds.
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We further develop the research of McCann-Pass on optimal transportation in unequal dimensions.