Immersed surface in a 4-manifold that can no be homotoped to an embedded surface

Question

I am just learning the basics of 4-manifold topology and I am requesting an example to put in my example cabinet. I would like an explicit example of a mooth immersion $f : \Sigma \to M$ where $\Sigma$ is a closed compact orientable surface and $M$ is a smooth compact closed 4-manifold such that $f$ can not be homotoped to an embedding. I gather that we can always homotope $f$ such that each point in the image has at most 2 preimages so it would be nice if the example where already in this form.

Answer 1

Consider the genus function on $M$, $g:H_2(M;\Bbb Z) \to \Bbb N$ with $g(\alpha)$ is the mininum genus of an embedded orientable surface representing the homology class $\alpha$. There are various ways (which all give very similar results) to construct lower bounds on $g(\alpha)$ for a fixed class coming from Donaldson theory, Seiberg-Witten theory and Heegard Floer homology. The proofs of these bounds are all rather delicate.

For a concrete example, there is a very famous theorem of Kronheimer and Mrowka (https://en.wikipedia.org/wiki/Thom_conjecture) that says in $\Bbb CP^2$ projective algebraic curves realize the minimal genus. In particular by the genus-degree formula (https://en.wikipedia.org/wiki/Genus%E2%80%93degree_formula), the genus of a smooth degree 3 curve is 1. It is not so hard to see that there is an immersion of $S^2 \to \Bbb CP^2$ with the same homology class as smooth degree 3 curve (take the union of 3 complex projective lines and tube them together). By choosing the 3 lines generically, you can guarantee the immersion has no self-intersection points which aren't transverse double points. By the Thom conjecture, there is no embedded sphere representing this homology class. There's a more thorough discussion of surfaces in $\Bbb CP^2$ in the beginning of Gompf and Stipsicz.