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Localized intersection of currents and the Lefschetz coincidence point theorem. (English) Zbl 1370.14005

This is a very interesting article. The authors study the localization problem for currents on manifolds and coincidences of maps on manifolds. The notion of a Thom class of a current is introduced and the localized intersection of two currents defined. By considering the intersection of a current with a submanifold the authors obtain a residue theorem. As an application the study of the existence of coincidence points of two maps is presented. In particular the global (and in case of a compact source manifold) local coincidence class of two maps is defined. The obtained results, a general coincidence theorem, which provides a formula describing the global coincidence class in terms of local classes, generalizes the celebrated Lefschetz coincidence theorem. Some other issues are also of independent interest. For instance explicit formulas for the coincidence point index are very neat. The paper is very well organized and clearly written. In the reviewer opinion it is accessible also for a not that well-experienced reader. The paper is an interesting contribution to the geometry of manifolds and the fixed point theory.

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
32C30 Integration on analytic sets and spaces, currents
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
55M05 Duality in algebraic topology
57R20 Characteristic classes and numbers in differential topology
58A25 Currents in global analysis
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References:

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