Algebraic topology of Polish spaces. II: Axiomatic homology

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 $\lim \text{colim}=\text{colim}\lim$.

24 pages, 1 figure

References in corpus (1)

• On the Axiomatic Systems of Steenrod Homology Theory of Compact Spaces

Cited by in corpus (1)

• Lim colim versus colim lim. II: Derived limits over a pospace