Coherent orientation of mixed moduli spaces in Morse-Floer theory
The goal of the paper under review is to provide a coherent orientation for the spaces of intersections of gradient trajectories and holomorphic disks in the cotangent bundle case along the line first introduced by \textit{S. Donaldson} [``The orientation of Yang-Mills moduli spaces and 4-manifold topology'', J. Differ. Geom. 26, 397--428 (1987; Zbl 0683.57005)] for moduli spaces of (anti)-self-dual connections and by \textit{A. Floer} and \textit{H. Hofer} [``Coherent orientations for periodic orbit problems in symplectic geometry'', Math. Z. 212, No.~1, 13--38 (1993; Zbl 0789.58022)] for the symplectic moduli spaces. The coherent orientation for Lagrangian intersections was given by Fukaya, Oh, Ohta and Ono and for Morse theory by Schwarz. Consider the cotangent bundle \(P=T^*M\) of a smooth compact closed \(n\)-dimensional manifold and a compactly supported Hamiltonian \(H: P \times [0, 1]\to\mathbb{R}\). Define \(L_0 = 0_M\) to be the zero section of \(P\) and \(L_1 = \phi_1^H(L_0)\) to be a Hamiltonian deformation of \(L_0\) by the symplectic diffeomorphism \(\phi_1\) generated by \(H\). Let \(CF_*(H)\) be the Floer chain groups generated by the transversal intersection points of \(L_0\) and \(L_1\) with \({\mathbb{Z}}_2\)-coefficients. The boundary map \(\partial_F: CF_*(H) \to CF_*(H)\) measures the number of solutions \(u: \mathbb R\times [0, 1]\to T^*M\) of \[ \frac{\partial u}{\partial s} + J\left (\frac{\partial u}{\partial t} - X_H(u)\right) = 0,\;u(s, i)\in L_i,\;i=0, 1,\;u(\pm \infty, t)\in L_0 \cap L_1. \] The authors constructed in [``Piunikhin-Salamon-Schwarz isomorphisms for Lagrangian intersections'', Differ. Geom. Appl. 22, No.~2, 215--227 (2005; Zbl 1064.37041)] an isomorphism between the Floer homology \(HF_*(H)\) of \(CF_*(H)\) and the Morse homology \(HM_*(f)\) of a Morse function \(f\) by connecting the gradient flow on \((-\infty, 0]\) with a \(J\)-holomorphic curve \(u\) to form a moduli space which gives rise to a chain homomorphism from \(CF_*(H)\) to \(CM_*(f)\) and induces an isomorphism on the homology level with \({\mathbb{Z}}_2\)-coefficients. For the cotangent bundle case, no holomorphic spheres or disks bubbles off with boundary in the zero section \(L_0\). The goal of the paper is to extend the \({\mathbb{Z}}_2\)-isomorphism to a \(\mathbb{Z}\)-isomorphism once the \(\mathbb{Z}\)-coefficients Floer chain groups are defined. For a point \(p\in M\) and a Hamiltonian path \(x(t)\) in \(T^*M\), beginning and ending in the zero-section \(0_M\), the authors consider spaces of pairs of maps (\(\gamma, u)\), \(\gamma:(-\infty,0]\to M\) and \(u:[0,\infty)\times[0,1]\to T^*M\), satisfying certain differential equations and boundary conditions, among them \(\gamma(0)=u(0,\frac{1}{2})\), i.e., they match with each other at end pints, where \(M\) is identified with \(0_M\). A class \(\Sigma^F\) of operators between Sobolev- and Lebesgue-spaces consisting of such pairs is considered. There is a gluing construction for Fredholm operators in the class \(\Sigma^F\). The idea is to glue the mixed-type Fredholm operators from \(\Sigma^F\) to \(\Sigma^M\), where \(\Sigma^M\) is the class of Fredholm operators arising from the linearized operators of the Morse gradient flow. Proposition 7 shows that the determinant line bundle of mixed-type gluing is also the tensor of the determinant line bundles from the Floer and the Morse case. Section 3 defines a coherent orientation on those non-parametric mixed moduli spaces which connects the Morse gradient flow with the J-holomorphic curve on the cotangent bundle, and shows the consistency of coherent orientations among the mixed moduli spaces, gradient flows and \(J\)-curves. In the last section, the authors construct a canonical orientation, compare it with the coherent one, and they define the characteristic sign. In this way, the Floer chain group and the chain homotopy relation are extended from \({\mathbb{Z}}_2\)-coefficients to \(\mathbb{Z}\)-coefficients, and the Morse homology of \(M\) is naturally isomorphic to the Floer homology of the cotangent bundle \(T^*M\) with \(\mathbb{Z}\)-coefficients.
scientific article; zbMATH DE number 5615325
scientific article