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Let A1 be a stratified space and suppose that both the complement of the n-skeleton and the «-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 (« — l)-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 «-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 «-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. Let A be a stratified space and suppose X is locally triangulable. The problems that we wish to study in this paper are those of the existence and classification of piecewise linear (PL) structures on A which are compatible in a sense made more precise below, with the stratification of A. This paper is a continuation of our earlier papers [2] and [3], but may be read independently of them. We recall some definitions. Let A be a space. A stratification of A is an increasing family of closed subsets of X, {X{n)\n > 1} such that A(_l) = 0, there exists a positive integer A such that X(N) = A, and for every n, each component of A(n) A(n_1) is open in X(n) A("_1). The space A(n) (respectively, A<n) A(""") is called the n-skeleton (respectively, n-stratum) of {A(n)}; while A is called the formal dimension of X. If A(n) — A"*"-1' is a (possibly empty) topological (TOP) w-manifold without boundary, then {Xw} is called a TOP stratification of A. A TOP stratified space is a pair (A, {A(n)}) where {A(n)} is a TOP stratification of A. When no confusion will arise, we suppress explicit mention of the stratification and denote a stratified space simply by A. Although the individual strata of a TOP stratified space are homogeneous when Received by the editors June 20, 1979. AMS (MOS) subject classifications (1970). Primary 57C25; Secondary 57C10.

Extending combinatorial piecewise linear structures on stratified spaces. II