Fukaya category of infinite-type surfaces (Q6568749)

Let \(M\)\ be a symplectic surface. If \(M\)\ is of finite type, its Fukaya category is well understood. \textit{D. Auroux} and \textit{I. Smith} [Forum Math. Sigma 9, Paper No. e26, 50 p. (2021; Zbl 1471.53070)] took the definition of the Fukaya category of infinite type surfaces to be a colimit of that of finite type surfaces by utilizing the operations of a pair of pants decomposition and gluing operations.\N\NRecently infinite type Riemann surfaces have attracted much interest in the study of big mapping class groups [\textit{D. Calegari}, ``Big Mapping Class Groups and Complex Dynamics'', Preprint, \url{https://math.uchicago.edu/~dannyc/courses/dynamics_2019/big_mcg_dynamics.pdf}; \textit{D. Calegari} and \textit{L. Chen}, Trans. Am. Math. Soc., Ser. B 9, 957--976 (2022; Zbl 1525.57005); Ergodic Theory Dyn. Syst. 41, No. 7, 1961--1987 (2021; Zbl 1470.37066)]. In particular, Calegari proposed to study the mapping class group \(\mathrm{Map}(\mathbb{R}^{2}\backslash C)\), where \(C\) denotes a Cantor set, and questioned whether the group has an infinite-dimensional space of quasimorphisms, as is the case with the mapping class group of a surface of finite topological type.\N\NMotivated by the study of big mapping class groups and Calegari's above proposal, this paper provides a geometric construction of a Fukaya category of infinite type surfaces by a direct construction without taking the colimit. This geometric construction and the study of Lagrangian spectral invariants is expected to yield a new dynamical approach to Calegari's question on the space of quasimorphisms for big mapping class groups. It should be mentioned that Ueda and his collaborators gave a scintillating construction of modular forms via the study of the Fukaya category of the divisor complements and the mirror symmetry of elliptic curves, in which the study of the Fukaya category of surfaces of infinite type naturally arise as some universal covering of punctured elliptic curves on which the mapping class group of the latter surface acts, see [\textit{A. Nagano} and \textit{K. Ueda}, Hokkaido Math. J. 51, No. 2, 275--286 (2022; Zbl 1502.14089); \textit{K. Hashimoto} and \textit{K. Ueda}, Proc. Am. Math. Soc. 150, No. 2, 547--558 (2022; Zbl 1483.14061)].\N\N\N\begin{itemize}\N\item[Part 1] summarizes \textit{I. Richards}' classification result [Trans. Am. Math. Soc. 106, 259--269 (1963; Zbl 0156.22203)], explaining the scheme of a construction of surfaces. The way to define the Fukaya category using the gradient sectional Lagrangian branes is founded.\N\N\item[Part 2] gives a construction of the aforementioned Fukaya category.\N\N\item[Part 3] describes generators and algebraic structure of the Fukaya category.\N\end{itemize}