Ends of maps and applications

Suppose e: M—• X is a map from a finite dimensional manifold to a locally compact space. Homotopy conditions on e are developed under which it can be extended to a proper map et M' —• X by adding boundary to M. This is applied in several settings to obtain geometric conclusions from homotopy data. For example: an embedding X C M of a manifold factor in a manifold of codimension > 3 is locally flat if and only if the complement M — X is locally 1connected at points of X. A completion of a map e: M —> X is a manifold M D M, with Mt M C bM', and an extension e: M —-> M which is proper. A map is proper if preimages of compact sets are compact. MAIN THEOREM (THE END THEOREM [11]). Suppose X is a locally compact locally 1-connected metric space. If M is a manifold of dimension > 6, e: M —• Xis a map which is proper on bM, and the end of e is onto, tame, and l'LC, then e has a completion. We define the terms used in the theorem. A neighborhood of the end is an open set UCM such that e\M U is proper. The end of e is onto if e(U) = X for every neighborhood U of the end. The end of e is 1-LC if for every x E X and neighborhoods U C M of the end, and V C X of x, there are neighborhoods if C U, V C V such that points (loops) in if d e~(V) can be joined by by arcs (resp. are nullhomotopic) in U O e~~(V). The end of e is tame if there are homotopies pulling M back from the end which are small in X. More precisely, for every e > 0 and neighborhood U of the end there is a neighborhood of the end V C U and a homotopy H: M x I —> M such that H0 = IM, HX(M) CM~V, Ht(M~ U)CM~V,md for each m G M the diameter of the arc eH: {m} x ƒ —> X is less than e. Tameness is a necessary condition for a completion, since such homotopies can be obtained by pushing M' in a collar of M' M C bM. If X is a point, this theorem is exactly the theorem of Browder, Livesay, and Levine [1], later extended to nonsimply connected ends by Siebenmann [12]. The proof in outline is similar to the classical X = pt. case. Homotopy and Received by the editors August 9, 1978. AMS (MOS) subject classifications (1970). Primary 57A99, 54C55, 57B05. 1 Partially supported by the National Science Foundation. ©American Mathematical Society 1979 0002-9904/7 9/0000-001S /$01.7 5