Spectral Floer theory and tangential structures

Abstract

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $\hat{I}˜\to \hat{I}¦$ of spaces living over $BO\to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $\hat{I}˜\to \hat{I}¦$. The previous case corresponded to $\hat{I}˜=\hat{I}¦=\{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $\hat{I}˜$ and $\hat{I}¦$ 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 $\hat{I}{\text{c◯}}_{\ast }^{(\hat{I}˜,\hat{I}¦),\circ }$. 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.

Comments welcome! v3: Accepted version, incorporates referee's comments and suggestions