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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 $$P$$.

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