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Analytic formulas for topological degree of non-smooth mappings: the even-dimensional case. (English) Zbl 1271.47047

The degree of a continuous map \(f: X\to Y\) between the topological spaces \(X\) and \(Y\) is one of the fundamental invariants in algebraic topology. Roughly speaking, such a degree measures the cardinality of pre-image of the map \(f\). The homology theory allows to express the degree of \(f\) in terms of determinant of the linear map induced by \(f\) on the homology groups. It is interesting, however, to have an “analytic” formula for the degree, i.e., a formula giving the degree of \(f\) in terms of index theory of the pseudo-differential operators on \(X\) and \(Y\).

This very interesting paper solves the problem for non-smooth mappings of the even-dimensional manifolds; the case of the odd-dimensional manifolds has been settled by the same author in [Adv. Math. 231, No. 1, 357–377 (2012; Zbl 1246.32034)]. The paper is clearly written and an example of \(X=S^{2n}\) of the even-dimensional sphere is considered in great detail. A draft of the paper is available at {http://arxiv.org/abs/1006.3954}.

MSC:

47H11 Degree theory for nonlinear operators
55M25 Degree, winding number
58J40 Pseudodifferential and Fourier integral operators on manifolds

Citations:

Zbl 1246.32034
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References:

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