Journal of Symplectic Geometry

Volume 17 (2019)

Number 3

Thin compactifications and relative fundamental classes

Pages: 703 – 752



Eleny-Nicoleta Ionel (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Thomas H. Parker (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)


We define a notion of relative fundamental class that applies to moduli spaces in gauge theory and in symplectic Gromov–Witten theory. For universal moduli spaces over a parameter space, the relative fundamental class specifies an element of the Čech homology of the compactification of each fiber; it is defined if the compactification is “thin” in the sense that the boundary of the generic fiber has homological codimension at least two.

The research of E.I. was partially supported by the Simons Foundation Fellowship #340899.

Received 1 November 2017

Accepted 25 July 2018

Published 9 September 2019