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arXiv · 1808.10243

Algebraic topology of Polish spaces. II: Axiomatic homology

Sergey A. Melikhov
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Abstract

Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common generalization of Milnor's two additivity axioms suffice to uniquely characterize (co)homology of closed pairs of Polish spaces (=separable complete metrizable spaces). The proof provides a combinatorial description of the (co)homology of a Polish space in terms of a cellular (co)chain complex satisfying a symmetry of the form $\lim \text{colim} = \text{colim} \lim$.

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Sergey A. Melikhov. 2018-08-30. Algebraic topology of Polish spaces. II: Axiomatic homology. https://arxiv.org/abs/1808.10243

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