Polytope Novikov homology (Q2239375)

For a closed smooth oriented and connected finite dimensional manifold \(M\), Sergey P. Novikov associated a homology with each a cohomology class \(a\in H^1_\mathrm{dR}(M)\), the so-called Novikov homology \(HN_\ast(a)\), cf. [\textit{S. P. Novikov}, Sov. Math., Dokl. 24, 222--226 (1981; Zbl 0505.58011); translation from Dokl. Akad. Nauk SSSR 260, 31--35 (1981), Russ. Math. Surv. 37, No. 5, 1--56 (1982; Zbl 0571.58011); translation from Usp. Mat. Nauk 37, No. 5(227), 3--49 (1982)]. Let \(\Phi_a:\pi_1(M)\to\mathbb{R}\) be the period homomorphism, and let \(\pi:\widetilde{M}_a\to M\) be the minimal regular covering with the group of deck transformations \(\Gamma_a\cong\pi_1(M)/\mathrm{Ker}(\Phi_a)\). Then for any representative \(\alpha\in a\) there exists an \(\tilde{f}_\alpha\in C^\infty(\widetilde{M}_a)\) such that \(\pi^\ast\alpha=d\tilde{f}_\alpha\). For a Riemannian metric \(g\) on \(M\) the pair \((\alpha, g)\) is said to be Morse-Smale if \((\tilde{f}_\alpha, \pi^\ast g)\) satisfies the Morse-Smale condition on \(\widetilde{M}_a\). For each \(i\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}\) let \(\mathrm{Crit}_i(\tilde{f}_\alpha)\) denote the critical points of \(\tilde{f}_\alpha\) with Morse index \(i\). The \(i\)th Novikov chain group \(\mathrm{CN}_i(\alpha)\) consists of all formal sums \[ \xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle \] such that \(\{\tilde{x}\mid \xi_{\tilde{x}}\in\mathbb{Z}\setminus\{0\}\,\&\, \tilde{f}_\alpha(\tilde{x})>c\}\) is finite for each \(c\in\mathbb{R}\). The boundary operator \(\partial : \mathrm{CN}_i(\alpha) \to \mathrm{CN}_{i-1}(\alpha)\) is defined by \[ \partial \xi:=\sum_{\tilde{x}, \, \tilde{y}} \, \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha}) \, \tilde{y}, \] where \(\#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha})\) counts trajectories of negative gradient of \(\tilde{f}_\alpha\) with respect to \(\tilde{g}:=\pi^\ast g\) with signs from \(\tilde{x}\) to \(\tilde{y}\). The Novikov ring \(\Lambda_\alpha\) consists of all formal sums \[ \lambda=\sum_{A\in\Gamma_a}\lambda_A A\in \bigoplus_{A\in\Gamma_a}\mathbb{Z}\langle A\rangle \] such that \(\{A\in\Gamma_a\mid \lambda_A\in\mathbb{Z}\setminus\{0\}\,\&\, \Phi_a(A)<c\}\) is finite for each \(c\in\mathbb{R}\). The product is given by the convolution \[ (\lambda\ast\mu)_A=\sum_{B\in\Gamma_a}\lambda_B\mu_{B^{-1}A}. \] According to the obvious action of \(\Lambda_a\) on \(\mathrm{CN}_\ast(\alpha)\), the latter is a finitely generated \(\Lambda_a\)-module. Moreover the boundary operator \(\partial\) is \(\Lambda_a\)-linear, and for each \(i \in \mathbb{N}_0\) the Novikov homology \[ \mathrm{HN}_i(\alpha,g):=\frac{\ker \partial_i}{\mathrm{im} \, \partial_{i+1}} \] carries a \(\Lambda_a\)-module structure. Different choices of cohomologous Morse forms representing \(\alpha\) induce isomorphic Novikov homologies. The isomorphism class is said to be the Novikov homology of pairs \((\alpha, g)\), and denoted by \(\mathrm{HN}_\ast(a)\). In the paper under review the author generalizes the above Novikov homology and defines polytope Novikov homology. Corresponding to a polytope \(\mathcal{A}=\langle a_0, \dots, a_k \rangle \subset H^1_{\mathrm{dR}}(M)\) with vertices \(a_0,\dots,a_k\), there exists a regular cover \(\pi : \widetilde{M}_{\mathcal{A}} \to M\) with the group of deck transformations \[ \Gamma_\mathcal{A}\cong {\pi_1(M)}{\bigg /}\bigcap_{l=0}^k \mathrm{Ker}(\Phi_{a_l}), \] Then for every \(a \in \mathcal{A}\) and for any representative \(\alpha\in a\) there exists a \(\tilde{f}_{\alpha} \in C^{\infty}(\widetilde{M}_{\mathcal{A}})\) such that \(\pi^*\alpha=d\tilde{f}_{\alpha}\). Fix a smooth section \(\theta : \mathcal{A} \longrightarrow \Omega^1(M)\), that is, \(\theta_a\) is a representative of \(a\). For each \(i\in\mathbb{N}_0\) the \(i\)th polytope Novikov chain complex group \(\mathrm{CN}_i(\theta_a,\mathcal{A})\) consists of all formal sums \[ \xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_{\theta_a})}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle \] such that \begin{gather*} \xi=\sum_{\tilde{x} \in \mathrm{Crit}_i\left(\tilde{f}_{\theta_a}\right)} \xi_{\tilde{x}} \, \tilde{x} \in \mathrm{CN}_i(\theta_a, \mathcal{A}) \iff \forall b \in \mathcal{A}, \forall c \in \mathbb{R} : \\ \#\lbrace \tilde{x} \mid \xi_{\tilde{x}} \neq 0, \; \tilde{f}_\beta(\tilde{x})>c \rbrace < +\infty, \end{gather*} where \(\beta \in b\) is any representative. The groups \(\mathrm{CN}_\bullet(\theta_a,\mathcal{A})\) may be equipped with boundary operators \(\partial_{\theta_a} : \mathrm{CN}_\ast(\theta_a,\mathcal{A}) \to \mathrm{CN}_{\ast-1}(\theta_a,\mathcal{A})\) given by \[ \partial_{\theta_a} \xi:= \sum_{\tilde{x}, \tilde{y}} \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}\left(\tilde{x},\tilde{y};\tilde{f}_{\theta_a}\right) \, \tilde{y}. \] Let \(\widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\) denote the upward completion of the group ring \(\mathbb{Z}[\Gamma_{\mathcal{A}}]\) with respect to the period homomorphism \(\Phi_b : \Gamma_{\mathcal{A}} \to \mathbb{R}\). Define the polytope Novikov ring \(\Lambda_\mathcal{A}=\bigcap_{b \in \mathcal{A}} \widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\). The above boundary operator \(\partial_{\theta_a}\) is \(\Lambda_{\mathcal{A}}\)-linear. The homology of the chain complex \(\left(\mathrm{CN}_\ast(\vartheta_a,g_{\vartheta_a},\mathcal{A}),\partial \right)\), denoted by \(\mathrm{HN}_\ast(\vartheta_a,\mathcal{A})\), is called the polytope Novikov homology. It is proved that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. An important application is to present a novel approach to the (twisted) Novikov Morse Homology Theorem: For any cohomology class \(a \in H^1_{\mathrm{dR}}(M)\) there exists an isomorphism \(\mathrm{HN}_\ast (a) \cong \mathrm{H}_\ast(M,\Lambda_a)\) of Novikov-modules. The second application is to prove a new polytope Novikov Principle, which generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case [\textit{A. Pajitnov}, Eur. J. Math. 6, No. 4, 1303--1341 (2020; Zbl 1470.57050)].