Disk-like capped gropes of height one and one-half

Use the framed four-dimensional block and stage conventions for gropes in the cited definitions. A capped grope is a staged thickening whose bottom surface stage is joined along curves in a symplectic basis to higher surface stages, with the final tip circles capped by framed immersed disks. A disk-like capped grope has one connected bottom surface stage with exactly one attaching circle. Its height is the number of successive surface stages on each side of the symplectic basis, with caps terminating the stages. A disk-like capped grope has height one and one-half, written $1.5$, when its two sides have heights $(1,0)$ or $(0,1)$ in the asymmetric convention; equivalently, one half of the first-stage basis has one additional surface stage and the other half is capped immediately. The attaching region is the framed solid-torus neighborhood of the single attaching circle. For an immersed disk, a double-point loop is the image of a path in the domain disk joining the two preimages of a transverse self-intersection. In the disk-embedding property for an input capped grope, this loop is read as the corresponding element of the grope's fundamental group; choosing a basing path from the grope basepoint fixes that based element.

Stefan Behrens et al., The Disc Embedding Theorem, Definitions 12.3, 12.5–12.6, and 12.10–12.12, https://dokumen.pub/the-disc-embedding-theorem-0198841310-9780198841319.html