The PL Grassmannian and PL curvature
HTML articles powered by AMS MathViewer
- by Norman Levitt PDF
- Trans. Amer. Math. Soc. 248 (1979), 191-205 Request permission
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.References
- Edgar H. Brown Jr., Cohomology theories, Ann. of Math. (2) 75 (1962), 467–484. MR 138104, DOI 10.2307/1970209
- C. P. Rourke and B. J. Sanderson, Block bundles. I, Ann. of Math. (2) 87 (1968), 1–28. MR 226645, DOI 10.2307/1970591
- David A. Stone, Sectional curvature in piecewise linear manifolds, Bull. Amer. Math. Soc. 79 (1973), 1060–1063. MR 320974, DOI 10.1090/S0002-9904-1973-13331-2
- David A. Stone, Geodesics in piecewise linear manifolds, Trans. Amer. Math. Soc. 215 (1976), 1–44. MR 402648, DOI 10.1090/S0002-9947-1976-0402648-8
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- André Haefliger and Valentin Poenaru, La classification des immersions combinatoires, Inst. Hautes Études Sci. Publ. Math. 23 (1964), 75–91 (French). MR 172296
- Norman Levitt and Colin Rourke, The existence of combinatorial formulae for characteristic classes, Trans. Amer. Math. Soc. 239 (1978), 391–397. MR 494134, DOI 10.1090/S0002-9947-1978-0494134-6
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 248 (1979), 191-205
- MSC: Primary 57Q99; Secondary 57R65
- DOI: https://doi.org/10.1090/S0002-9947-1979-0521700-2
- MathSciNet review: 521700