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Simplicial volumes of Alexandrov spaces. (English) Zbl 0914.53028
This paper extends M. Gromov’s results [Publ. Math., Inst. Hautes Étud. Sci. 56, 5-99 (1982; Zbl 0516.53046)] to any \(n\)-dimensional singular space (that is, a locally compact space \(X\) with singular space \(S\) such that \(X-S\) is a topological \(n\)-manifold and \(S\) has a lower dimension \(<n\)). Let \(X\) be a compact orientable \(n\)-dimensional singular space having a fundamental class \([X]\). Some sufficient conditions satisfied by a length metric of \(X\) are given here such that the Hausdorff \(n\)-measure coincides with the mass \(([X])\) used in Gromov’s paper cited above.
An upper (respectively lower) bound for the Hausdorff \(n\)-measure of certain \(n\)-dimensional compact orientable Alexandrov spaces with curvature \(K\leq -1\) (respectively \(K\geq -1\)) is given here by using the simplicial volume. The simplicial volume of a singular space \(X\), denoted by \(\| X\|\), having a fundamental class, is defined by \[ \| X\|=\inf_{[X]}\| [X]\|_1, \] where \(\|[X]\|_1\) denotes the \(\ell^1\)-norm of \([X]\), and \([X]\) runs over all the fundamental classes of \(X\).
Reviewer: C.-L.Bejan (Iaşi)

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55U15 Chain complexes in algebraic topology
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