Localized intersection of currents and the Lefschetz coincidence point theorem

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• 作者：Cinzia Bisi ; Filippo Bracci ; Takeshi Izawa…
• 关键词：Alexander duality ; Thom class ; Localized intersections ; Residue theorem ; Coincidence classes and indices ; Lefschetz Coincidence point formula
• 刊名：Annali di Matematica Pura ed Applicata
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：195
• 期：2
• 页码：601-621
• 全文大小：535 KB
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  8.Lehmann, D.: Systèmes d’alvéoles et intégration sur le complexe de Čech-de Rham, Publications de l’IRMA, vol. 23, N\({}^{o}\) VI. Université de Lille I, Villeneuve-d’Ascq (1991)
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  14.Vick, J.W.: Homology Theory: An Introduction to Algebraic Topology, Graduate Texts in Math. Springer, Berlin (1994)CrossRef MATH
  
• 作者单位：Cinzia Bisi (1)
  Filippo Bracci (2)
  Takeshi Izawa (3)
  Tatsuo Suwa (4)
  
  1. Dipartimento di Matematica ed Informatica, Università di Ferrara, Via Machiavelli, n. 35, 44121, Ferrara, Italy
  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, n. 1, 00133, Rome, Italy
  3. Department of Information and Computer Science, Hokkaido University of Science, Sapporo, 006-8585, Japan
  4. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1618-1891

文摘

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular, we consider the situation where we have a \(C^\infty \) map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Čech–de Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.