Do the virtual fundamental classes satisfy functorial properties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:45:03Z http://mathoverflow.net/feeds/question/30111 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30111/do-the-virtual-fundamental-classes-satisfy-functorial-properties Do the virtual fundamental classes satisfy functorial properties? HYYY 2010-06-30T22:02:37Z 2010-07-06T08:43:14Z <p>In Gromov–Witten theory, if the symplectic virtual fundamental classes constructed by B.Siebert satisfy functorial properties, i.e., if $f\colon X\to Y$ is an appropriate map between symplectic manifolds $X$ and $Y$, then <code>$f_*\colon [X]^{\rm vir}=[Y]^{\rm vir}$</code>? In his paper constructing symplectic GW invariant, I didn't see he mentions this, so does anyone knows anything about this? Thanks!</p> http://mathoverflow.net/questions/30111/do-the-virtual-fundamental-classes-satisfy-functorial-properties/30740#30740 Answer by Barbara for Do the virtual fundamental classes satisfy functorial properties? Barbara 2010-07-06T08:43:14Z 2010-07-06T08:43:14Z <p>This may be not the answer you want, but in algebraic geometry there are such results, particularly in the context mentioned in Kevin Lin's comment. They usually apply to virtual classes constructed from relative, not absolute, obstruction theories. Two of them I know of are a Lemma of Kevin Costello and (particularly pertinent for Lin's comment) Cristina Manolache's applications of her own virtual pullbacks. </p>