A new proof of Brown's collaring theorem

Robert Connelly
1971 Proceedings of the American Mathematical Society  
The aim of this note is to give a new proof that if a subspace B, compact for convenience, is locally collared in a space X, then it is collared. The idea of the proof is simply to add a collar BXI to X to get X+ and then to construct a homeomorphism of X with X+ by pushing B down on one collared open set at a time. The theorem, of course, is essentially that of [l]. However, the proof easily works in the piecewise linear (PL) category (i.e. all maps are PL and spaces are polyhedra), and when
more » ... yhedra), and when B, the boundary, is a pair or flag, cf. [3] . At the end of the paper we shall note briefly how the noncompact case and the PL case can be handled by our techniques. A closed subspace BEX is locally collared if B is covered by sets U, open in B, such that for each U there is a closed embedding A:Z7X[0, 1]->X such that h~\B) =Vx{0}, ä(x, 0)=x for xEU, and hiUX [O, 1)) is open in X. For metric spaces this is equivalent to the definition in [l]. B is said to be collared if one U can be taken to be all of B. Theorem. If BEX is compact and locally collared in X, which is Hausdorff, then B is collared in X.
doi:10.1090/s0002-9939-1971-0267588-7 fatcat:pdegev2e2na53hmlfjg3v4qb6a