Naturality of Floer homology of open subsets in Lagrangian intersection theory (Q2752848)

The author proves naturality of the (filtered) Floer homology \(HF^\lambda_* (H, U: M)\) constructed by \textit{R. Kasturirangan} and the author [Math. Z. 236, 151-189 (2001; Zbl 0985.53039)] in cotangent bundles with conormal boundary condition for an open set \(U\) in the base manifold \(M\), where \(H:[0, 1]\times T^*M\to\mathbb R\) is a Hamiltonian and \(\lambda\) is a real number. That is, for any given pair \(U\subset\overline U\subset V\) with smooth boundaries there exists a canonical homomorphism NEWLINE\[NEWLINE\iota^H_{UV}: HF_*^\lambda (H, U: M)\to HF_*^\lambda(H, V: M)NEWLINE\]NEWLINE commuting with the canonical isomorphism \(F_{H,U}: H_*(U, {\mathbb Z})\to HF_*^{\lambda}(H, U: M)\) with respect to the homology homomorphism \(i_*: H_*(U, {\mathbb Z}) \to H_*(V, {\mathbb Z})\). Then the author defines the ``Čech''-version of the Floer cohomology for an arbitrary locally closed subset \(C\subset M\), NEWLINE\[NEWLINE\check HF^*_\lambda(H, C: M):= \varinjlim_{U\subset C}HF^*_\lambda(H, U: M)NEWLINE\]NEWLINE and then proves that there exists a canonical isomorphism \(\check F_{(H,C)}: \check H^*(C, {\mathbb Z})\to \check F^*_\infty(H, C: M)\). On these results, the author sets the stage of quantization of Eilenberg-Steenrod Axioms.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].