Non-orientable genus of knots in punctured Spin 4-manifolds

TL;DR

This paper proves that for any Spin 4-manifold, the minimal first Betti number of certain non-orientable surfaces bounded by knots in the boundary can grow arbitrarily large, generalizing known results about the non-orientable 4-ball genus.

Contribution

It establishes that the non-orientable genus function $ ext{γ}_X^0$ is unbounded for all Spin 4-manifolds, extending previous results from the 4-sphere case.

Findings

$ ext{γ}_X^0$ is unbounded for any Spin 4-manifold

Generalizes Batson's 2012 result for $S^4$

Provides new insights into non-orientable surface genera in 4-manifolds

Abstract

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $\gamma_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $\gamma^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $\gamma^0_X$ is a generalization of the non-orientable 4-ball genus. While it is very likely that for given $X$, $\gamma^0_X$ has no upper bound, it is difficult to show it. In fact, even in the case of $\gamma^0_{S^4}$, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that for any Spin 4-manifold $X$, $\gamma^0_X$ has no upper bound.