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Homology and cohomology theories on manifolds. (English) Zbl 1226.57035
The authors study some generalized homology and cohomology theories on the category of smooth, PL, and topological manifolds. These are, by definition, absolute theories satisfying a Mayer-Vietoris property. They find a functor $$F$$ which assigns to a generalized (co-)homology theory $$h_*$$ on manifolds a theory $$\hat{h}_*$$ on the category $$\mathcal{C}$$ of countable CW complexes of finite dimension and they show that this functor is full and faithful, and that that $$F$$ is a bijection on morphism sets. After presenting some well known facts about CW structures in the section 2, they obtain a generalization to the case of triples in section 3. After presenting some axioms for ordinary (co-)homology on manifolds, the authors give, for certain coefficient rings, an axiomatic characterization of the cup product on the singular cohomology on manifolds.

##### MSC:
 57N65 Algebraic topology of manifolds 57R19 Algebraic topology on manifolds and differential topology 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology