Nonstandard homology theory for uniform spaces
| Autor: | Takuma Imamura |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Pure mathematics Cellular homology 010102 general mathematics Homology (mathematics) 01 natural sciences Mathematics::Geometric Topology Mathematics::Algebraic Topology CW complex 010101 applied mathematics 55N35 54J05 Morse homology Mayer–Vietoris sequence Mathematics::K-Theory and Homology FOS: Mathematics Moore space (algebraic topology) Algebraic Topology (math.AT) Geometry and Topology Mathematics - Algebraic Topology 0101 mathematics Mathematics::Symplectic Geometry Singular homology Relative homology Mathematics |
| Popis: | We introduce a new homology theory of uniform spaces, provisionally called $\mu$-homology theory. Our homology theory is based on hyperfinite chains of microsimplices. This idea is due to McCord. We prove that $\mu$-homology theory satisfies the Eilenberg-Steenrod axioms. The characterization of chain-connectedness in terms of $\mu$-homology is provided. We also introduce the notion of S-homotopy, which is weaker than uniform homotopy. We prove that $\mu$-homology theory satisfies the S-homotopy axiom, and that every uniform space can be S-deformation retracted to a dense subset. It follows that for every uniform space $X$ and any dense subset $A$ of $X$, $X$ and $A$ have the same $\mu$-homology. We briefly discuss the difference and similarity between $\mu$-homology and McCord homology. Comment: Accepted by Topology and its Applications; corrigendum added |
| Databáze: | OpenAIRE |
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