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Local homology and dimensional full-valuedness. (English. Russian original) Zbl 1149.55001
Math. Notes 81, No. 5, 573-589 (2007); translation from Mat. Zametki 81, No. 5, 643-659 (2007).
The spaces in this paper are locally compact, metrizable, and finite dimensional with respect to covering dimension dim. For such a space $$X$$, $$\text{hdim}_{\mathbb Z} X$$ denotes the homological dimension of $$X$$ with respect to $$\mathbb Z$$. This equals the largest number $$n$$ for which $$H_n(X,X\setminus\{x\};\mathbb Z)\neq0$$ for some $$x\in X$$.
The interest is in the question of which spaces $$X$$ have the property that $$\text{dim}(X\times Y)=\text{dim}X+\text{dim}Y$$ for all $$Y$$, i.e., the logarithmic law holds for all $$Y$$. Such a space $$X$$ is called dimensionally full-valued. Not all spaces have this property, so one may ask for criteria on $$X$$ to guarantee it.
The author remarks that in [A. E. Kharlap, Mat. Sb., N. Ser. 96(138), 347–373 (1975; Zbl 0312.55006)] it was noticed that homology manifolds over the ring $$\mathbb Z$$ are dimensionally full-valued. Also it is pointed out that under the assumption of homological local connectedness, the $$(\mathbb Z-n)$$-spaces in [G. E. Bredon, Sheaf Theory, Springer-Verlag, New York (1997; Zbl 0874.55001)] are dimensionally full-valued. In this paper it is proved that homological local connectedness, even with an additional condition called peripheral homological connectedness, does not ensure dimensional full-valuedness.
A quasipolyhedron is a space whose local homology is finitely generated. It is highest local homology is torsion free, then it is called a homological polyhedron. Let us state a theorem from Section 1.
Theorem: A quasipolyhedron $$X$$ is dimensionally full-valued if and only if it is a homological polyhedron. In this case, $$h\text{dim}_{\mathbb Z}X=\text{dim}_{\mathbb Z}X=\text{dim}X$$.

MSC:
 55M10 Dimension theory in algebraic topology 54F45 Dimension theory in general topology
Citations:
Zbl 0312.55006; Zbl 0874.55001
Full Text:
References:
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