The 0-concordance monoid admits an infinite linearly independent set

Abstract

Under the relation of $0$-concordance, the set of knotted 2-spheres in $S^4$ forms a commutative monoid $\mathcal{M}_0$ with the operation of connected sum. Sunukjian has recently shown that $\mathcal{M}_0$ contains a submonoid isomorphic to $\mathbb{Z}^{\ge 0}$. In this note, we show that $\mathcal{M}_0$ contains a submonoid isomorphic to $(\mathbb{Z}^{\ge 0})^\infty$. Our argument relates the $0$-concordance monoid to linear independence of certain Seifert solids in the (spin) rational homology cobordism group.