Cobordism and Concordance of Surfaces in 4-Manifolds

TL;DR

This work classifies cobordism and concordance of surfaces embedded in simply-connected 4-manifolds, extending prior results to arbitrary X and to nonorientable cases through Pin$^\pm$-surgery and spanning-manifold techniques. The main theorems tie cobordism to ${\mathbb{Z}/2}$-homology and normal Euler-number data, and show that, in the simply-connected setting, cobordant surfaces are concordant when the necessary Euler-number and boundary conditions align. The authors develop a robust framework of spanning manifolds, Seifert sections, and relative Euler-number theory, enabling precise control of boundaries, extensions, and ambient surgeries. A key methodological advance is the unoriented ambient surgery program, built on Pin$^\pm$-structures, which allows converting cobordisms into concordances in codimension 2 within a simply-connected ambient. The results unify smooth and topological categories and yield a complete concordance classification for closed surfaces in simply-connected 4-manifolds, with broad implications for the topology of embedded surfaces in 4-manifolds.

Abstract

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are $\mathbb{Z}/2$-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the 4-manifold is simply-connected and the surfaces are closed, non-orientable, and cobordant, we show that they are in fact concordant. This completes the classification of closed surfaces in simply-connected 4-manifolds up to concordance. Our methods give new constructions of cobordisms with prescribed boundaries, and completely determine when a given cobordism between the boundaries extends to a cobordism or concordance between the surfaces. We obtain our concordance results by extending Sunukjian's method of ambient surgery to the unoriented case using Pin$^-$-structures. We also discuss conditions for an arbitrary codimension 2 properly embedded submanifold to admit an unoriented spanning manifold with prescribed boundary. All results hold in both the smooth and topological categories.

Figures (7)

• Figure 1: Example of input data for Theorem \ref{['mainthm: general cobordism']}, with reduced dimensions. Here $X=D^2$, $\Sigma_0 \cong I \sqcup I$, $\Sigma_1 \cong I$, and $Z \cong I \sqcup I \sqcup I$. In this example, $Z$ extends to a cobordism from $\Sigma_0$ to $\Sigma_1$ (not shown).
• Figure 2: A proper embedding with $X = D^3$ and $\Sigma \cong I \sqcup I$ (red), and spanning surface $Y$ (yellow) extending $Z \cong I \sqcup I$ (green).
• Figure 3: A proper codimension 2 embedding with $X = D^3$ and $\Sigma = I \sqcup I$ (red). A spanning surface $Y$ (yellow) is given. The push-off $\Sigma^s_+$ of $\Sigma$ determined by the Seifert section associated to $Y$ is shown (dashed red).
• Figure 4: A simple model, demonstrating that $[\gamma^*s(S^1)] \cdot [u(S^1)] = [s(B)] \cdot [\alpha(S^1)]$. Left: the $S^1$-bundle $M$, above $B$; a curve $\alpha$ (red), and projection $\gamma$ (dashed); the images of two sections $s,s'\colon B \to M$ (green,blue). The top and bottom faces of $M$ are identified. Right: the $S^1$-bundle $\gamma^*M$, above $S^1$; the curve $u$ (red); the images of the two sections $\gamma^*s,\gamma^*s'$ (green, blue). Note that the intersections between $\alpha$ and $s$ (resp. $s'$) in $M$ correspond to the intersections between $u$ and $\gamma^*s$ (resp. $\gamma^*s'$) in $\gamma^*M$.
• Figure 5: A diagrammatic representation of how to convert all double points into type (I). Top: example converting from a double point of type (IV) to one of type (III), as well as two of type (I). Bottom: example converting from a pair of double points of type (III) to six of type (I).

Theorems & Definitions (120)

• Theorem A
• Corollary B
• Theorem C
• Theorem D
• Theorem E
• Corollary F
• Theorem 1: ThomQuelqueProprietes
• Theorem 2
• proof : Proof of Theorem \ref{['mainthm: concordance rel boundary']}
• Theorem 3