Relative smooth triangulations, strong approximation and bundle fitting

A finite geometric simplicial complex $K$ is considered with its Euclidean realization and induced metric; subcomplex, closure, star and link terminology follows Subcomplexes, closures, stars, and links in a simplicial complex. For $b\in \mathrm{\mid }K\mathrm{\mid }$ the tangent star is the union of the tangent cones at $b$ of all simplices containing $b$, that is, the rays locally contained in the closed star.

A map $f:\mathrm{\mid }K\mathrm{\mid }\to M$ is piecewise smooth when every restriction $f\mathrm{\mid }\sigma$ to a closed simplex extends smoothly to a neighbourhood in its affine hull and the extensions agree on common faces. In a target coordinate chart the differential $d{f}_b$ is then the well-defined continuous piecewise linear map on the tangent star that equals $D(f\mathrm{\mid }\sigma )(b)$ on the cone of each simplex $\sigma$ containing $b$; agreement on common faces makes it independent of the chosen containing simplex. A smooth immersion of a complex requires $d{f}_b$ injective on the entire tangent star at every point, not merely on each simplex tangent space; a smooth embedding requires in addition that $f$ be a homeomorphism onto its image, and a finite smooth triangulation of $M$ is such an embedding onto $M$. A corner-adapted triangulation must be face compatible: the inverse image of each local corner face is locally the restriction of a subcomplex, and the differential fan maps the source tangent star homeomorphically onto the appropriate tangent orthant, or its specified face sector. This last local orthant condition is the corner-adapted triangulation contract.

After an ambient Euclidean realization of the target, or in a fixed finite family of compact target chart patches, $g:K^{\mathrm{\prime }}\to M$ on a finite subdivision $K^{\mathrm{\prime }}$ of $K$ is a strong $C^1$ $\delta$-approximation to $f$, for $\delta >0$, if ${\sup }_b\mathrm{\mid }g(b)-f(b)\mathrm{\mid }<\delta$ and $\mathrm{\mid }d{g}_b(v)-d{f}_b(v)\mathrm{\mid }<\delta \mathrm{\mid }v\mathrm{\mid }$ for all $b$ and all nonzero tangent-star vectors $v$. A relative approximation fixed on $L$ means $g{\mathrm{\mid }}_{\mathrm{\mid }L\mathrm{\mid }}=f{\mathrm{\mid }}_{\mathrm{\mid }L\mathrm{\mid }}$, subdivision of $L$ permitted unless explicitly prohibited. Shape control means a positive lower bound on $r(\sigma )\mathrm{/}\mathrm{diam}\sigma$ for every positive-dimensional simplex, where $r(\sigma )$ is the distance from the barycentre to the relative boundary; it is not asserted for arbitrary iterated barycentric subdivisions.

Two embeddings $f_i:K_i\to M$ fit in a subcomplex if the inverse images of $f_1(\mathrm{\mid }{K}_1\mathrm{\mid })\cap f_2(\mathrm{\mid }{K}_2\mathrm{\mid })$ are subcomplexes $L_i$ and $f_2^{-1}{f}_1:L_1\to L_2$ is a simplexwise affine isomorphism; the fit is full if every simplex all of whose vertices belong to $L_i$ belongs to $L_i$. For a full fit, gluing along that isomorphism gives a finite simplicial complex and its union map: simplices from opposite sides can share only vertices of the glued full subcomplex, so their common vertices span a common face. For a fit that is not full, first barycentrically subdivide both sides compatibly to make the common subcomplex full, then glue. The union map is well defined because the maps agree on the identified points.

For a smooth bundle $\pi :E\to B$ and a supplied base triangulation $h:D\to B$, a projection-preserving diagram consists of $F:K\to E$ and a map $r:K\to D$ that is affine on each simplex into a simplex of the original $D$ and satisfies $\pi F=hr$; the affine/PL requirement does not require domain vertices to map to vertices of the original fixed base complex. Fitting preserves the projection when all modified maps satisfy the same equality with the original $h$ and the original affine $r$ restricted to their subdivisions; a fixed relative bundle piece must already carry such a diagram. A projection-preserving diagram is determined by the pair $(F,r)$ together with its restrictions to subdivisions, so a modification supported away from a piece leaves the original diagram there exactly on the nose. No generic smooth map is called a bundle merely because it is proper.

For relative extension on a compact manifold with corners, supplied boundary data are admissible only when they are compatible on all corner-face intersections and satisfy the tangent-orthant condition above; the extension keeps their pointwise parametrization, with compatible subdivisions permitted, and no extension of arbitrary incompatible boundary data is asserted.