Image nonconcordance of positive-genus ${\pi }_1$-injective surfaces

Abstract

We construct, for every $g\ge 2$, infinite families of homotopic smooth embeddings of a closed genus-$g$ surface whose images are pairwise not smoothly image-concordant, while each surface is ${\pi }_1$-injective. The main closed examples lie in one-fold stabilizations of closed aspherical mapping tori with torsion-free fundamental group: after stabilization by $S^2\times S^2$, the surfaces have a common framed dual sphere and the inclusion of each complement induces a ${\pi }_1$-isomorphism. The image-nonconcordance already occurs before stabilization, in the underlying closed aspherical mapping torus, and persists after every finite number of $S^2\times S^2$-stabilizations. The obstruction is a computable mod-two coordinate of Freedman--Quinn/Dax-type self-intersection data for concordance tracks, indexed by self-dual double-cosets of a possibly non-normal surface subgroup $H\le {\pi }_{1}X$. The geometric source of the relevant labels is a M"obius-band square-root relation: elements $t\notin H$ with $t^2\in H$ produce self-dual labels in torsion-free ambient groups. These square roots are realized naturally in Klein-bottle $I$-bundle pieces and persist in closed graph-manifold mapping-torus examples.

55 pages, 3 figures. Corrected an error in Section 2.6 and improved the exposition