2024/11/05 by Porcelli, Noah, Smith, Ivan · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2411.03257
In \citePS, for a stably framed Liouville manifold X we defined a Donaldson-Fukaya category F(X;\mathbbS) over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from F(X;ℤ) to F(X;\mathbbS). Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' Θ→ Φ of spaces living over BO → BU, whose objects are Lagrangians L→ X for which the classifying maps of their tangent bundles lift to Θ→ Φ. The previous case corresponded to Θ= Φ= \pt\. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of Θ and Φ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory Ω(Θ,Φ),∘_*. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum R should exist, which may be of independent interest.