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Homology of weighted simplicial complexes. (English) Zbl 0735.18011

The author generalizes the notion of a simplicial complex to that of a weighted simplicial complex. He then constructs an ”integral” homology functor \(H\) defined on this category and proves that, with a suitable notion of contiguity, \(H\) satisfies the Eilenberg-Steenrod axioms. He further shows that \(H\) differs from homology for standard simplicial complexes by providing a simple example for which the positive- dimensional homology is zero but which has torsion in dimension zero. He explains the apparent paradox in categorical terms. Finally he applies his constructions to provide a homology functor on the category of preconvexity spaces.

MSC:

18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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References:

[1] 1 R. J. MacG. Dawson , A simplification of the Eilenberg-Steenrod axioms for the category of finite simplicial complexes , J. Pure & Appl. Algebra 53 ( 1988 ), 257 - 265 . MR 961363 | Zbl 0659.55004 · Zbl 0659.55004 · doi:10.1016/0022-4049(88)90126-0
[2] 2 R. J. MacG. Dawson , Limits and colimits of preconvexity spaces , Cahiers de Topologie et Géométrie Differentielle XXVIII ( 1987 ), 307 - 328 . Numdam | MR 926549 | Zbl 0637.52001 · Zbl 0637.52001
[3] 3 R. J. MacG. Dawson , Homology of preconvexity spaces , Cahlers de Topologie et Géométrie Différentielle , to appear. · Zbl 0795.52001
[4] 4 Eilenberg & Steenrod , Foundations of algebraic Topology , Princeton 1952 . Zbl 0047.41402 · Zbl 0047.41402
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