Journal of Symplectic Geometry

Volume 19 (2021)

Number 2

Polyfold regularization of constrained moduli spaces

Pages: 241 – 350



Benjamin Filippenko (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


We introduce tame sc‑Fredholm sections and slices of sc-Fredholm sections. A slice is a notion of subpolyfold that is compatible with the sc‑Fredholm section and has finite locally constant codimension. We prove that the subspace of a tame polyfold that satisfies a transverse sc-smooth constraint in a finite dimensional smooth manifold is a slice of any tame sc‑Fredholm section compatible with the constraint. Moreover, we prove that a sc‑Fredholm section restricted to a slice is a tame sc-Fredholm section with a drop in Fredholm index given by the codimension of the slice. As a corollary, we obtain fiber products of tame sc‑Fredholm sections. We describe applications to Gromov–Witten invariants, constructing the Piunikhin–Salamon–Schwarz maps for general closed symplectic manifolds, and avoiding sphere bubbles in moduli spaces of expected dimension $0$ and $1$.

This material is based upon work supported by the National Science Foundation under Award No. 1708916 and No. 1903023.

Received 26 July 2018

Accepted 30 July 2020

Published 27 May 2021