The PL Grassmannian and PL curvature

Abstract:

A space ${\mathcal {G}_{n,k}}$ is constructed, together with a block bundle over it, which is analogous to the Grassmannian ${G_{n,k}}$ in that, given a PL manifold ${M^n}$ as a subcomplex of an affine triangulation of ${R^{n + k}}$, there is a natural “Gauss map” ${M^n} \to {\mathcal {G}_{n,k}}$ covered by a block-bundle map of the PL tubular neighborhood of ${M^n}$ to the block bundle over ${G_{n,k}}$. Certain subcomplexes of ${G_{n,k}}$ are then studied in connection with immersion problems, the chief result being that a connected manifold ${M^n}$ (nonclosed) PL immerses in ${R^{n + k}}$ satisfying certain “local” conditions if and only if its stable normal bundle is represented by a map to the subcomplex of ${G_{n,k}}$ corresponding to the condition. An important example of such a condition is a restriction on PL curvature, e.g., nonnegative or nonpositive, PL curvature having been defined by D. Stone.