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.

10 pages; final version. Published in Proc. Amer. Math. Soc