Lefschetz formulas for flows.

*(English)*Zbl 0619.58034
Differential equations, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, Pt. III, Contemp. Math. 58, No. 3, 19-69 (1987).

[For the entire collection see Zbl 0607.00012.]

According to the known Wilson plug construction every nonsingular flow \(\phi\) may be deformed through nonsingular flows to a flow \(\psi\) with no closed orbits. So there are no correlations between the existence of closed orbits of the nonsingular flow \(\phi\) and the topology of the underlying manifold M. The author shows, nonetheless, that for many types of \(\phi\) there is a quantitative relation between closed orbits and the underlying topology of M. This relation the author calls Lefschetz formula for \(\phi\). Namely: let E be a flat vector bundle over M, \(\phi_ E(\gamma)\) holonomy of E around the loop \(\gamma\). Then zeta function may be introduced as: \[ Z_{Q,E}(S)=\prod_{\gamma}(\det I- \Delta (\gamma)\rho_ E(\gamma)e^{-Sl(\gamma)})^{(-1)^{u(\gamma)}} \] where \(\gamma\) runs over all prime closed orbits of \(\phi\), u(\(\gamma)\) is the dimension of the unstable bundle over \(\gamma\), \(\Delta (\gamma)=\pm 1\) as this bundle is orientable or not. Let us note, also, that \(\psi_{\phi,E}=| Z_{\phi,E}(0)|\) and \(\tau\) (E) is l-torsion (Reidemeister torsion). Then according to the author the pair (\(\phi\),E) is called Lefschetz if \(\psi_{\phi,E}=\tau (E)\). One of the main statements of the article is the following: ”If the chain recurrent set R(\(\phi)\) is circular (i.e. admits cross-sections) then for certain bundles E, (\(\phi\),E) is Lefschetz” (§ 3). Some other results may be mentioned: if X has constant positive curvature, then any acyclic bundle over X is Lefschetz (cor. 6.2). For oriented X of constant negative curvature, any acyclic unitary E over SX is Lefschetz (Theorem 6.5, where SX is a unit sphere bundle of X). At the end of the paper the author conjectures that: if X is a compact locally homogeneous Riemannian manifold and E is an acyclic bundle over SX then E is Lefschetz.

According to the known Wilson plug construction every nonsingular flow \(\phi\) may be deformed through nonsingular flows to a flow \(\psi\) with no closed orbits. So there are no correlations between the existence of closed orbits of the nonsingular flow \(\phi\) and the topology of the underlying manifold M. The author shows, nonetheless, that for many types of \(\phi\) there is a quantitative relation between closed orbits and the underlying topology of M. This relation the author calls Lefschetz formula for \(\phi\). Namely: let E be a flat vector bundle over M, \(\phi_ E(\gamma)\) holonomy of E around the loop \(\gamma\). Then zeta function may be introduced as: \[ Z_{Q,E}(S)=\prod_{\gamma}(\det I- \Delta (\gamma)\rho_ E(\gamma)e^{-Sl(\gamma)})^{(-1)^{u(\gamma)}} \] where \(\gamma\) runs over all prime closed orbits of \(\phi\), u(\(\gamma)\) is the dimension of the unstable bundle over \(\gamma\), \(\Delta (\gamma)=\pm 1\) as this bundle is orientable or not. Let us note, also, that \(\psi_{\phi,E}=| Z_{\phi,E}(0)|\) and \(\tau\) (E) is l-torsion (Reidemeister torsion). Then according to the author the pair (\(\phi\),E) is called Lefschetz if \(\psi_{\phi,E}=\tau (E)\). One of the main statements of the article is the following: ”If the chain recurrent set R(\(\phi)\) is circular (i.e. admits cross-sections) then for certain bundles E, (\(\phi\),E) is Lefschetz” (§ 3). Some other results may be mentioned: if X has constant positive curvature, then any acyclic bundle over X is Lefschetz (cor. 6.2). For oriented X of constant negative curvature, any acyclic unitary E over SX is Lefschetz (Theorem 6.5, where SX is a unit sphere bundle of X). At the end of the paper the author conjectures that: if X is a compact locally homogeneous Riemannian manifold and E is an acyclic bundle over SX then E is Lefschetz.

Reviewer: V.B.Marenich