Extending combinatorial piecewise linear structures on stratified spaces. II

Anderson, Douglas R.; Hsiang, Wu Chung

doi:10.1090/S0002-9947-1980-0570787-8

Extending combinatorial piecewise linear structures on stratified spaces. II
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by Douglas R. Anderson and Wu Chung Hsiang

Trans. Amer. Math. Soc. 260 (1980), 223-253

DOI: https://doi.org/10.1090/S0002-9947-1980-0570787-8

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Abstract:

Let X be a stratified space and suppose that both the complement of the n-skeleton and the n-stratum have been endowed with combinatorial piecewise linear (PL) structures. In this paper we investigate the problem of “fitting together” these separately given PL structures to obtain a single combinatorial PL structure on the complement of the $(n - 1)$-skeleton. The first main result of this paper reduces the geometrically given “fitting together” problem to a standard kind of obstruction theory problem. This is accomplished by introducing a tangent bundle for the n-stratum and using immersion theory to show that the “fitting together” problem is equivalent to reducing the structure group of the tangent bundle of the n-stratum to an appropriate group of PL homeomorphisms. The second main theorem describes a method for computing the homotopy groups arising in the obstruction theory problem via spectral sequence methods. In some cases, the spectral sequences involved are fairly small and the first few differentials are described. This paper is an outgrowth of earlier work by the authors on this problem.

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Seminar on combinatorial topology