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, \(h\text{dim}_{\mathbb Z}X=\text{dim}_{\mathbb Z}X=\text{dim}X\).

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, \(h\text{dim}_{\mathbb Z}X=\text{dim}_{\mathbb Z}X=\text{dim}X\).

Reviewer: Leonard R. Rubin (Norman)

##### Keywords:

dimension; dimensionally full-valued; homological dimension; homological manifold; homological polyhedron; local homology; locally compact space; peripherally homologically locally connected; quasipolyhedron
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\textit{D. V. Artamonov}, Math. Notes 81, No. 5, 573--589 (2007; Zbl 1149.55001); translation from Mat. Zametki 81, No. 5, 643--659 (2007)

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