# Asymptotic Geometry – Relationship between Gromov-Witten and the Gromov invariant of Taubes

Fix a symplectic and compact four collector ($$X$$, $$omega$$).

Recall the Gromov invariant of Taubes is a certain full value function on $$H ^ 2 (X; mathbb {Z})$$ defined by the weighted number of pseudoholomorphic curves in $$X$$. In particular, this number combines curves of all kinds.

On the other hand, the Gromov-Witten invariants for a fixed genus $$g$$ and homology course $$A in H_2 (X; mathbb {Z})$$ are (very roughly) integers derived from the "fundamental class" of module space $$mathcal {M} _g ^ A (X)$$ of pseudoholomorphic maps of a kind surface $$g$$ in $$X$$ representing the class $$A$$.

Is there a relationship, conjectural or otherwise, between these two invariants stronger than "both have holomorphic curves"? Of course, this would require looking at the Gromov-Witten invariants for all genres $$g$$. Personally, I do not understand the best way to define them for the genre $$g> 0$$. I understand that Zinger has a construction for $$g = 1$$ it's a bit more refined than looking at all the modules space.