Index classes for vector fields and their relation to certain characteristic classes (Q2847665)

The author introduces cohomology classes to express the indices of a vector field \(V\) on a smooth, compact, oriented \(m\)-dimensional manifold \(M\) with boundary \(\partial M\). The vector field \(V\) has isolated zeros and has no zeros on the boundary of \(M\). Each zero has an index and their sum is the index of \(V\) denoted by \(I^\circ(V)\). For a closed manifold (empty boundary), \(I^\circ(V) = \chi(M)\), the Euler-Poincaré number. This is called the Poincaré-Hopf Theorem. In 1929, Marston Morse proved a theorem which determines the index of \(V\) for manifolds with boundary, NEWLINE\[NEWLINEI^\circ(V) + I^{-}(V) = {\chi (M)}NEWLINE\]NEWLINE Here \(I^{-}(V)\) is the index of the vector field restricted to the boundary and projected to a vector field tangent to that part of the boundary where \(V\) is pointing inwards. Morse actually used the index \(I^{+}(V)\) in the formula which introduced a change in sign depending on the dimension \(m\) of the manifold. \(I^{+}(V)\) is the part of the vector field on the boundary which points inside, projected down to the surface of the boundary. Morse's theorem was not applied or widely known until 1968 when \textit{C. C. Pugh} published [A generalized Poincaré index formula, Topology 7, 217--226 (1968; Zbl 0194.24602)]. It was rediscovered again in 1986 by the reviewer in the form given above.NEWLINENEWLINEThe author defines three cohomology classes: \(u^\circ\), \(u^-\), \(u^+\) in \(H^m(TM,TM_{0}^{\partial})\). Here, \(TM\) is the tangent bundle of \(M\) and \(TM_{0}^{\partial}\) is the tangent bundle restricted to the boundary of \(M\) with the zero cross-section removed.NEWLINENEWLINEHere \(v\) is the cross-section to the Tangent bundle which corresponds to the vector bundle \(V\), and \(v^*\) is the induced homomorphism of \(v\) in cohomology. Also \(\mu\) is the fundamental class of \((M, \partial M)\). NEWLINE\[NEWLINEI^{\circ}(V) = \langle v^*u^\circ, \mu \rangle NEWLINE\]NEWLINE NEWLINE\[NEWLINE I^{-}(V) = \langle v^*u^-, \mu \rangleNEWLINE\]NEWLINE NEWLINE\[NEWLINE I^{+}(V) = \langle v^*u^+, \mu \rangleNEWLINE\]NEWLINE The actual definitions of \(u^\circ\), \(u^-\), \(u^+\) are variations of Thom classes given in complicated commutative diagrams. They simplify to NEWLINE\[NEWLINEu^\circ := kuNEWLINE\]NEWLINE where \(k\) is an inclusion of pairs of tangent bundles and \(u\) is the Thom class. And NEWLINE\[NEWLINEu^- := -\delta'_s (\delta_s)^{-1} ju^\circNEWLINE\]NEWLINE And NEWLINE\[NEWLINEu^+ :=\delta' (\delta)^{-1} ju^\circNEWLINE\]NEWLINE where \(\delta\) is a connection homomorphism and \(s\) is an inward pointing vector field.NEWLINENEWLINE\textit{Ji-Ping Sha} in [Ann. Math. (2) 150, No. 3, 1151--1158 (1999; Zbl 0980.57012)] defined a secondary Chern-Euler class, \(\Upsilon \epsilon H^{m-1}(TM_0^\partial ; Z[1/2])\). Then \(2\Upsilon={\tilde{u}}^+ - \tilde{u}^-\) where \(\tilde{u}\) is a unique preimage of \(u\).NEWLINENEWLINEAlso \textit{V. A. Šarafutdinov} [Sib. Math. J. 14(1973), 930--940 (1974; Zbl 0299.55022)] defined a relative Euler class \(e(E,n)\). Let \(n\) be a outwards pointing vector field on the boundary of \(M\) to \(TM\), then NEWLINE\[NEWLINE\langle e(TM, n), \mu \rangle = \chi(M)NEWLINE\]