Behavior of genus function on a 4-manifold for sums

Question

Let $X$ be a smooth compact 4-manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \mathbb{Z}) \to \mathbb{Z}_{\geq 0}$ which assigns to a homology class the smallest genus of such a smooth surface needed to represent it. Suppose that $x$ is a nontorision element of $H_2(X; \mathbb{Z})$. Does the sequence $G(x), G(2x), G(3x),...$ limit to infinity? Can there be arbitrarily large zeroes? Is there always a limit?

Answer 1

In the case that $x\cdot x \neq 0$, topological methods based on the G-signature show that the genus goes to infinity more or less quadratically in $n$. (I'll be more specific below.) This goes back to Rochlin (Two-dimensional submanifolds of four-dimensional manifolds) and Hsiang-Szczarba (On embedding surfaces in 4-manifolds) in the 1970s.

Following Rochlin's version (since I don't have the other at hand): if a homology class $\xi$ is divisible by $h$, an odd prime power, then $$ g \geq \left|\frac{(h^2-1)(\xi \cdot \xi)- \sigma(X)}{4 h^2}\right| - \frac{b_2(X)}{2}. $$ Writing $\xi = h \alpha$ we see that the right side grows quadratically in such $h$. (Generally this grows as the square of the largest prime power dividing $n$ where $\xi = n \alpha$; presumably the growth rate of that quantity in $n$ is known.)

By looking in a neighborhood (and sticking to prime powers), you can see that you'd expect quadratic growth, but the estimate above looks off by a factor of two. For instance, when it holds, the adjunction formula (as quoted by Marco above) gives a bound that is roughly twice the G-signature bound.

Work of Strle (Bounds on genus and geometric intersections from cylindrical end moduli spaces) gives stronger results for surfaces of positive self-intersection in the case that $b_2^+(X) =1$, without the assumption of non-vanishing Seiberg-Witten invariants. See also recent work of Konno (Bounds on genus and configurations of embedded surfaces in 4-manifolds).

Finally, in the case of self-intersection $0$, the growth is at most linear (and possibly $0$, as Marco notes). This follows by tubing together parallel copies of a given surface.

Answer 2

Sometimes the function $G$ can be constantly 0: consider the class $x = [S^2\times\{p\}]$ in $H_2(S^2\times F)$, where $F$ is a surface. Then $G(nx)$ can be realised by an embedded sphere for all $n$: just pick $n$ distinct points $p_1,\dots,p_n$ in $F$, and tube $S^2\times\{p_i\}$ to $S^2\times\{p_{i+1}\}$ (using pairwise disjoint tubes).

As for the existence of a limit, to me this is a lot less clear. Certainly something is known when $b^+(X) > 1$ and some Seiberg–Witten invariant of $X$ does not vanish, at least in the case when $x\cdot x > 0$. Then there is the adjunction inequality (Kronheimer–Mrowka), telling you that (for some second cohomology class $K$, corresponding to a non-vanishing SW invariant) $$ 2G(nx) - 2 \ge |\langle K, x\rangle| + n^2x\cdot x. $$ The right-hand side of the inequality grows quadratically, so $G(nx)$ goes to $\infty$.

I'd be very curious to know of "interesting" behaviours of the function $n \mapsto G(nx)$ (e.g. non-monotonicity, frequent non-monotonicity, eventual constant non-zero behaviour, periodicity/aperiodicity).