The PL Grassmannian and PL curvature

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A space G n , k {\mathcal {G}_{n,k}} is constructed, together with a block bundle over it, which is analogous to the Grassmannian G n , k {G_{n,k}} in that, given a PL manifold M n {M^n} as a subcomplex of an affine triangulation of R n + k {R^{n + k}} , there is a natural “Gauss map” M n → G n , k {M^n} \to {\mathcal {G}_{n,k}} covered by a block-bundle map of the PL tubular neighborhood of M n {M^n} to the block bundle over G n , k {G_{n,k}} . Certain subcomplexes of G n , k {G_{n,k}} are then studied in connection with immersion problems, the chief result being that a connected manifold M n {M^n} (nonclosed) PL immerses in R n + k {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 {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.