\(\eta\)-invariants and the Poincaré-Hopf index formula (Q2735685)

The paper provides an ``index theorem-method'' proof of the famous Poincaré-Hopf formula expressing the Euler characteristic of a smooth closed even-dimensional manifold \(M\) in terms of local data of isolated singular points of a vector field \(V\) on \(M\), namely NEWLINE\[NEWLINE\chi(M)= \sum_{x\in B(V)} \deg_V (x),NEWLINE\]NEWLINE where \(B(V)\) is the (finite) set of zeros of the vector field \(V\), and \(\deg_V(x)\) is the degree of the map from a ``small'' sphere with the center at \(x\) to the standard unit sphere in the Euclidean space determined by \(V\).NEWLINENEWLINENEWLINENow a brief sketch of the proof follows. First we equip the manifold \(M\) with a Riemannian metric \(g\), and assume that \(g\) is flat near any zero point of \(V\). In fact we can assume that for any \(x\in B(V)\) there is a ball \(B(x)\) with the center at \(x\) an open neighborhood of which is isometric to an open neighborhood of the standard unit ball in the Euclidean space. Now, let \(D=d+d^*: \Omega^* \to \Omega^*\) be the de Rham-Hodge operator on \(M\), and let \(D_{o/e}\) be the restriction of \(D\) to \(\Omega^{\text{odd/even}}\) respectively. Then we have NEWLINE\[NEWLINE\chi (M)=\text{index} D_e.NEWLINE\]NEWLINE The first step of the proof is a study of ``global'' elliptic boundary value problems of the type of Atiyah-Patodi-Singer. Let \({\mathcal M}=\) Closure \((M\setminus \cup_{x\in B(V)} B(x))\), \(D_{M,o/e}\) be the restriction of \(D_{o/e}\) to \({\mathcal M}\), \(D_{\partial M,o/e}\) -- the tangential part of \(D_{M,o/e}\), and let \(P_{\partial M,\geq 0,e}\) (resp. \(P_{\partial M,\geq 0,o}\), resp. \(P_{\partial M,>0,o})\) be the orthogonal projection from the \(L^2\)-completion of \(\Omega^{\text{even}}\mid\partial{\mathcal M}\) (resp. \(\Omega^{\text{odd}} \mid\partial {\mathcal M}\), resp. \(\Omega^{\text{odd}}\mid \partial{\mathcal M})\) to the direct sum of eigenspaces corresponding to non-negative (resp. non-negative, resp. positive) eigenvalues of \(D_{\partial M,e}\) (resp. \(D_{\partial M,o}\), resp. \(D_{\partial M,o})\). Then the elliptic boundary value problem \((D_{M,e}, P_{\partial M,\geq 0,e})\) is adjoint to \((D_{M,o}, P_{\partial M,>0, o})\). NEWLINENEWLINENEWLINENow the author, using Gilkey's generalization of the Atiyah-Patodi-Singer local index theorem for elliptic boundary value problems to the case when the metric is not of product nature and the flatness of the metric near \(B(V)\), shows that NEWLINE\[NEWLINE\chi(M)=\text{index}(D_{M,e}, P_{\partial M,\geq 0,e}).NEWLINE\]NEWLINE The next step of the proof consists in passing from the global formula above to a ``local'' formula expressing \(\chi(M)\) in terms of the operators restricted to a neighborhood of \(B(V)\), and the path from the global to the ``local'' formula is provided by a subtle correspondence between variations of elliptic boundary value problems and the spectral flow. First the author defines a variation \((D_{M,e},\widehat P_{\partial M,\geq o,e}=\) (by definition) \(\widehat c(V) P_{\partial M,\geq 0,o}\widehat c(V))\) of the elliptic boundary value problem \((D_{M,e}, P_{\partial M,\geq 0,e})\) (where \(\widehat c(V)\) is a suitable Clifford matrix) and observes that this new boundary value problem is also elliptic. Next, using a result of Gilkey, the author shows that \(\text{index} (D_{M,e}, P_{\partial M,\geq 0,e})= -\text{index} (D_{M,e},\widehat c(V)P_{ \partial M,\geq 0,o}\widehat c(V))\), and using a variation formula for variations of Dirac-type boundary value problems gets the following formula: NEWLINE\[NEWLINE\begin{aligned} \text{index} (D_{M,e},P_{\partial M,\geq 0,e}) & =1/2 \biggl(\text{index}(D_{M,e}, P_{\partial M,\geq 0,e})- \text{index} \bigl(D_{M,e}, \widehat c(V)P_{\partial M, \geq 0,o}\widehat c(V)\bigr) \biggr)\\ & =1/2 \text{index} T(P_{\partial M, \geq 0,e}, \widehat P_{\partial M,\geq 0,e}),\end{aligned}NEWLINE\]NEWLINE where \(T(P_{\partial M, \geq 0,e}, \widehat P_{\partial M,\geq 0,e})=P_{\partial M,\geq 0,e}\widehat P_{ \partial M,\geq 0,e}: \text{Im}(\widehat P_{\partial M,\geq o,e})\to \text{Im} (P_{\partial M,\geq 0,e})\) is a Fredholm operator. Now, the author considers a 1-parameter family \(D_{\partial M,e}(u) =(1-u)D_{\partial M,e} +u\widehat D_{ \partial M,e}\) of elliptic operators on \(\partial {\mathcal M}\), where \(\widehat D_{ \partial M,e}=\widehat c(V)D_{\partial M,o}\widehat c(V)\) and \(u\in[0,1]\). Since \(\widehat P_{\partial M,\geq 0,e}\) defines the Atiyah-Patodi-Singer boundary condition for \(\widehat D_{\partial M,e}\), the spectral flow \(SJ(\widehat D_{ \partial M,e}(u))-\text{index} T(P_{\partial M,\geq 0,e},\widehat P_{\partial M, \geq 0,e})\) by an earlier result of the author. Thus we arrive at the following formula: NEWLINE\[NEWLINE\chi(M)= 1/2 sf(D_{\partial M,e}(u): 0\leq u\leq 1).NEWLINE\]NEWLINE However \(\partial {\mathcal M}=-\bigcup_{x\in B(V)} \partial B(x)\), and \(D_{\partial B(x),e} (u)=\) the restriction of \(-D_{\partial M,e}(u)\) to \(\partial B(x)\) is just the path of Dirac-type operators corresponding to the standard Euclidean metric on the sphere \(\partial B(x)\). Thus finally we get the required ``local'' formula NEWLINE\[NEWLINE\chi(M)=-1/2 \sum_{x\in B(V)}sf \bigl(D_{\partial B(x),e}(u):0\leq u\leq 1\bigr).NEWLINE\]NEWLINE The remaining part of the proof consists in computing the spectral flow \(sf(D_{\partial B(x),e}(u): 0\leq u\leq 1)\). First the author expresses this spectral flow in terms of reduced eta-invariants and a heat kernel asymptotic. To be more precise, using a relation between spectral slow and reduced eta-invariants, the author gets the following formula: NEWLINE\[NEWLINEsf\bigl( D_{ \partial B(x),e}(u): 0\leq u\leq 1\bigr): \int^1_0{c_{-1/2}\over \sqrt\pi} du+ \overline\eta \bigl(D_{\partial B(x),e} (1)\bigr)-\overline \eta\bigl(D_{ \partial B(x),e} (0)\bigr),NEWLINE\]NEWLINE where \(c_{-1/2}\) appears as the coefficient of \(t^{-1/2}\) in an asymptotic expansion as \(t\to 0^+\) of \(\text{Tr} [{\partial \over \partial u}D_{\partial B(x),e} (u)\exp(-t(D_{\partial B(x),e}(u))^2)]\).NEWLINENEWLINENEWLINEAn easy inspection shows NEWLINE\[NEWLINE\overline\eta (D_{\partial B(x),e}(1))= \overline\eta (D_{\partial B(x),e}(0)),NEWLINE\]NEWLINE thus finally we get NEWLINE\[NEWLINEsf\bigl(D_{\partial B(x),e}(u): 0 \leq u\leq 1\bigr)= \int^1_0{c_{-1/2} \over\sqrt \pi}du.NEWLINE\]NEWLINE The final step of the proof is the computation (non-trivial) of the right-hand side of the last formula, which proves it to be equal to \(-2\deg_V(x)\). This, together with the formula above expressing \(\chi(M)\) in terms of the spectral flow, proves the Poincaré-Hopf formula.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].