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2020•arXiv (Cornell University)Open access

On a new geometric homology theory

Hao Yu

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Abstract

In this note we present a new homology theory, we call it geometric homology theory (or GHT for brevity). We prove that the homology groups of GHT are isomorphic to the singular homology groups, which solves a Conjecture of Voronov. GHT has several nice properties compared with singular homology, which makes itself more suitable than singular homology in some situations, especially in chain-level theories. We will develop further of this theory in our sequel paper.

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What this paper is about

In this note we present a new homology theory, we call it geometric homology theory (or GHT for brevity). We prove that the homology groups of GHT are isomorphic to the singular homology groups, which solves a Conjecture of Voronov. GHT has several nice properties compared with singular homology, which makes itself more suitable than singular homology in some situations, especially in chain-level theories. We will develop further of this theory in our sequel paper.

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Available abstract

In this note we present a new homology theory, we call it geometric homology theory (or GHT for brevity). We prove that the homology groups of GHT are isomorphic to the singular homology groups, which solves a Conjecture of Voronov. GHT has several nice properties compared with singular homology, which makes itself more suitable than singular homology in some situations, especially in chain-level theories. We will develop further of this theory in our sequel paper.

Key concepts: Cellular homology, Homology (biology), Relative homology, Morse homology, Singular homology, Mayer–Vietoris sequence, Conjecture, Mathematics

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