## Do the virtual fundamental classes satisfy functorial properties?

In Gromovâ€“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 $f_*\colon [X]^{\rm vir}=[Y]^{\rm vir}$? In his paper constructing symplectic GW invariant, I didn't see he mentions this, so does anyone knows anything about this? Thanks!

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In editing your question to use LaTeX for the formulas, I replaced an equals sign by an arrow. I hope this was correct. – Harald Hanche-Olsen Jun 30 2010 at 23:58
I suspect that was meant to be an equals sign, but that the notation is bad. The question seems to be asking two things (correct me if I'm wrong): 1) Does an "appropriate" $f$ induce a map on moduli spaces $f': M_{g,n}(X,\beta) \to M_{g,n}(Y,f_* \beta)$? And, 2) Is there a reasonable condition on $f$ under which the virtual dimensions coincide and $f'_* [M_{g,n}(X,\beta)]^{vir} = [M_{g,n}(Y,f_*\beta)]^{vir}$? – Anatoly Preygel Jul 1 2010 at 0:56
Okay, at least three people seem to think the equals sign was intended, so I put it back. I also kept the original colon. The formula makes no sense at all to me now, but then this is not my field, so I'll stay away from further editing and commenting now. – Harald Hanche-Olsen Jul 1 2010 at 13:17
Harald: $\overline{M}_{g,n}(X)$ is a certain moduli space of maps from curves to $X$. The virtual fundamental class is a certain (co)chain or (co)homology class on this moduli space. So, given a map $X \to Y$, we might naively expect a corresponding map $\overline{M}_{g,n}(X) \to \overline{M}_{g,n}(Y)$ gotten by sending $C \to X$ to the composition $C \to X \to Y$. Then one might ask whether this map is compatible with the two virtual fundamental classes via pullback or pushforward. – Kevin Lin Jul 1 2010 at 18:25
Mathphysics, what candidates do you have in mind for an "appropriate" map? The structure-preserving maps between symplectic manifolds are the symplectic immersions. But, for instance, the inclusion of a point doesn't preserve the virtual dimension of the space of stable maps. – Tim Perutz Jul 2 2010 at 20:19

## 1 Answer

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.

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