Homology Classes of Semi-Algebraic Sets and Mass Minimization

2014/04/18 by Quentin Funk, Funk, Quentin
Mathematics · #14P10 #49Q15 #55N35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:14P10 #msc:49Q15 #msc:55N35

paper · pdf · doi:10.48550/arxiv.1404.4796

16 pages, fixed typos, results unchanged

arxiv created 2014/11/06 · arxiv updated 2014/11/07

Abstract

We associate to any compact semi-algebraic set X ⊂ \mathbb Rⁿ a chain complex of currents S_∗ (X) generated by integration along semi-algebraic submanifolds and we analyze the corresponding homology groups. In particular, we show that these homology groups satisfy the Eilenberg-Steenrod axioms and further, that they are isomorphic to both the ordinary singular homology groups of X and to the homology groups generated by the integral currents supported on X. Using this result and a certain neighborhood of X, we are able to prove homological mass minimization for integral currents supported on X, and verify that any cycle of X that has sufficiently small mass is a boundary.

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