2025/09/25 by Porcelli, Noah, Smith, Ivan
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2509.21483
Let X be a graded Liouville domain. Fix a pair of infinite loop spaces Ψ= (Θ→ Φ) living over (BO → BU). This determines a spectral Fukaya category F(X;Ψ) whenever TX lifts to Φ, containing closed exact Lagrangians L for which TL lifts compatibly to Θ; and by Bott periodicity and index theory, a Thom spectrum R with bordism theory R_*. This paper has two main goals: we incorporate rank one spectral local systems ξ: L → BGL1(R) into the spectral category; and we prove that the bordism class [(L,ξ)] defined by the open-closed map differs from the class [L] by a multiplicative two-torsion element in R0(L)× determined by an action of the stable homotopy class of the Hopf map η∈ π1st on ξ. Methods include a twisting construction associating flow categories to spectral local systems, and a model for the open-closed map incorporating Schlichtkrull's construction of the trace map BGL1(R) ⊆ K(R) → R. The companion paper \citePS4 shows that (for Lagrangians which themselves admit spectral lifts) one can lift quasi-isomorphisms from ℤ to Ψ at the cost of introducing rank one local systems. Together with the open-closed computation given here, this gives an essentially complete picture of the bordism-theoretic consequences of quasi-isomorphism in the classical exact Fukaya category.