Whitney disk, clean Whitney disk and framed Whitney disk

Let $\gamma =\alpha \ast \beta$ be a Whitney circle for two complementary sheets, with distinct transverse corners $p,q$. Use the genuine two-corner bigon $\mathcal{B}=\{(u,v):-1\le u\le 1,\mathrm{\mid }v\mathrm{\mid }\le 1-u^2\}$ as its source. A Whitney disk is a map $W:\mathcal{B}\to X$ smooth as a map from a manifold with corners, meaning that it admits a smooth local extension near every source point, with the fixed product corner charts, taking its two boundary edges to $\alpha ,\beta$, with the prescribed product corner collars, and transverse to both sheets on its interior. A clean disk is embedded, its open interior misses the two sheets, and its boundary collars are adapted to the sheets: along each open edge its inward tangent is transverse to the corresponding sheet, so the disk tangent plane intersects the sheet tangent space in exactly the edge tangent line. An immersed Whitney disk has injective differential on the full two-dimensional tangent space at every source point, including boundary and corners, with the same fixed corner data, allowing interior self-intersections and intersections with the sheets.

For a clean or immersed disk, ${\nu }_W:=W^{\ast }TX\mathrm{/}dW(T\mathcal{B})$ is a rank-$(m-2)$ smooth normal bundle, using the full-rank local extensions at the boundary and corners. A general Whitney disk map may have singular interior points; no normal bundle is asserted for such a map. An admissible boundary frame is a pair of orthogonal partial frames of ranks $a-1,b-1$ as in Opposite local signs give the compatible Whitney-circle framing, adapted to the respective sheet collars and matched at the corners. A summand of rank zero has its unique empty frame; this convention includes the one-dimensional-sheet local model. The compatible-framing lemma supplies existence only in its stated positive-rank range. A clean framed Whitney disk is a clean disk equipped with a smooth normal trivialization extending an admissible boundary frame. The boundary frame is part of the data; the existence theorem permits an allowed one-summand correction before extension. A specified arbitrary full boundary frame is extendible exactly when its loop discrepancy from a disk frame is nullhomotopic. The rounded boundary circle has rank-$m-1$ normal bundle; its inward tangent-to-disk line is additional to the disk normal bundle. Null-homotopy, cleanliness and admissible frame extension are separate conditions.