A Hopf index theorem for a real vector bundle (Q1861026)

The main achievement of the paper is an ``index theory'' proof of an extension of the well-known Hopf index theorem (Euler characteristic of an oriented \(2n\) dimensional bundle \(E\) over \(2n\) dimensional smooth compact manifold \(X\) equals the sum of indexes of a transversal section \(\nu\) of \(E\) taken over all (isolated) zero points of \(\nu\)) to sections \(\nu\), which are non-degenerate in the sense of Bott. Here is a slightly more detailed presentation of the main result. First recall that a section \(\nu\) as above is said to be non-degenerate in the sense of Bott if its zero point consists of finitely many smooth oriented submanifolds \(V_k\), \(k=1,\dots,m\), of \(X\), and the natural map \(L_\nu : TX|V_k\to E|V_k\) (just the composition of the differential of the map \(\nu : X\to E\) and projection of \(TE|X\) onto the subbundle \(TE_{\text{vert}}\) of vertical vectors; note that there is a natural decomposition \(TE|X = TX\oplus TE_{\text{vert}}|X\), thus there is no ambiguity in definition of the projection) is injective on the subbundle \(N_k\), normal to \(V_k\) in \(X\). Now, let \(E_{Y, k} = (E|Y_k) / L_\nu(N_k)\). Then the main theorem of the paper states that for a section \(\nu\) as above \(\chi(E) = \sum^m_1\chi(E_{Y,k})\). In order to prove the theorem the authors define a suitably deformed (in the style of Witten) twisted Dirac operator on the manifolds \(Y^l_k\) (\(l\) is the dimension of \(Y^l_k\)) and prove that its index equals \((-1)^{l/2}2^{2n}\chi(E_{Y,k})\) if \(l\) is even and 0 if \(l\) is odd. Next the authors relate a suitable alternate sum of indexes of these Dirac operators to the index of a deformed Dirac-type operator on \(X\), which in turn is shown to be precisely \((-1)^n\chi(E)\) thus completing the proof.