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Elliptic equivalence of vector bundles. (English) Zbl 1041.58011
Let \(M\) be a smooth, connected, paracompact manifold and let \(E\), \(F\) be complex vector bundle over \(M\) of the same rank. The bundles are said to be elliptically equivalent (elliptically related) if one can find an elliptic pseudodifferential operator \(P:C^\infty(M,E)\rightarrow C^\infty(M,F)\). The authors look for the necessary and sufficient conditions for elliptic equivalence. If \(M\) is orientable, and either noncompact or odd dimensional then it is known that the elliptic equivalence implies that \(E\) and \(F\) are isomorphic. So they focus on orientable, compact and even dimensional manifolds. The answers are given in terms of the behaviour of Chern classes of \(E\) and \(F\) and the Euler class of \(M\). The method of constructing all vector bundles over \(M\) elliptically related to a given bundle is presented. Moreover it is proved that the index \(\text{ Ind}(P)=0\) if rank \(r\) of the operator \(P\) is smaller than \(m\), \(\text{ dim}M=2m\) and \(r=m\) and the Euler class of \(M\) is not zero than \( \text{ Ind}(P)\) depends on \(E\) and \(P\).

58J05 Elliptic equations on manifolds, general theory
58J40 Pseudodifferential and Fourier integral operators on manifolds
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