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Simple homotopy types and finite spaces. (English) Zbl 1146.57034
Kompakte Polyeder lassen sich bekanntlich durch (die Eckenmengen) endliche(r) Simplizialkomplexe beschreiben. Andererseits ergibt sich nach P. S. Alexandroff [Textbook of Set theory, Frankfurt/Main: Verlag Harri Deutsch (1994; Zbl 0833.04001)] auf einer teilweise geordneten endlichen Menge \(X\) eine Topologie: Als Basis nehme man Umgebungen der Form \(U_x= \{y\in X\mid y\leq x\}\). In der vorliegenden Arbeit werden diese Zuordnungen verknüpft, um (durch geeignete Begriffe) die Theorie des einfachen Homotopietyps von simplizialen Komplexen auf endliche Räume zu übertragen. Einem elementaren simplizialen Kollaps entspricht dabei z.B., dass ein (bezüglich der Eckenordnung so genannter “schwacher”) Punkt aus \(X\) entfernt wird.

MSC:
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
55U05 Abstract complexes in algebraic topology
06A06 Partial orders, general
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