Cohomology as a Local-to-global Bridge
نویسنده
چکیده
I discuss low dimensional incarnations of cohomology and illustrate how basic cohomological principles lead to a proof of Sperner’s lemma. CONTENTS 1. Chains and cochains with Z/2-coefficients 1 2. Application to Sperner Lemma 4 3. Where is the cohomology? 6 4. What next? 8 References 8 1. CHAINS AND COCHAINS WITH Z/2-COEFFICIENTS To keep the formalism at a minimum I will concentrate only on triangulated spaces of dimension ≤ 2. These are compact spaces equipped with a decomposition as a finite union of points vertices (or 0-simplices), edges (or 1-simplices) and triangles (or 2-simplices). Two edges can have in common at most one end-point, triangles can have in common only an edge or only a single vertex. An edge and a triangle can have in common either a single vertex, or the edge could be an entire edge of the triangle; see Figure 1.
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تاریخ انتشار 2009