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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The PL Grassmannian and PL curvature


Author: Norman Levitt
Journal: Trans. Amer. Math. Soc. 248 (1979), 191-205
MSC: Primary 57Q99; Secondary 57R65
MathSciNet review: 521700
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0521700-2
Keywords: Grassmannian, block-bundle, Gauss map, PL curvature, immersion
Article copyright: © Copyright 1979 American Mathematical Society