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Under special conditions, it is shown how to adjust topological concordances so as to leave certain sets fixed. Also, a lemma is given which establishes the existence of pinched collars and bicollars under specified circumstances. These two results are then combined to show how to obtain nice extensions of certain homeomorphisms. If X and Y are topological spaces and ho and h1 are embeddings of X into Y, a topological concordance between ho and h1 is an embedding H:XXI-*YXI (1= [O, 1]) such that H(x, 0)=(ho(x), 0) and H(x, 1) = (hi (x), 1) for all xE X. If A CX, a topological concordance H between ho:X-Y and h1:XY is fixed on A if H(x, t) = (ho(x), t) forall (x,t)ELAXI. The following theorem generalizes the principal result of [4], i.e., Theorem 4.1 of [4]. In fact, the following theorem reduces to Theorem 4.1 of [4], when M is a point. THEOREM. Let Mm be an m-dimensional PL-manifold and let Qq, q-m > 3, be a q-dimensional PL-manifold. Suppose that M is (2m-q+2)-connected, dOM is (2m-q+ 1)-connected (or empty), and Q is (m+ 1)-connected. Let f: M->Int Q be a locally fiat embedding. If ho and h1 are topological homeomorphisms of Q onto itself which leave f(M) fixed and which are topologically concordant, then there is a topological concordance between ho and h1 which isfixed onf (M). REMARK 1. The above theorem can be generalized in a straightforward way to permit f to be certain allowable embeddings. We leave the calculation of the necessary connectivity conditions to the reader. REMARK 2. A generalization of the piecewise linear version of the theorem has been obtained in a joint paper with L. S. Husch [6] since the submission of this paper. Also, the piecewise linear version Received by the editors March 28, 1969. AMS subject classifications. Primary 5702, 5705, 5478, 5560, 5570.

Adjustment of topological concordances and extensions of homeomorphisms over pinched collars