Embeddings with mapping cylinder neighborhoods

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Musr a submanifold Mm C N” having a mapping cylinder neighborhood be locally flat? This question has been answered in dimensions n ~4 and its relation with the infamous double suspension problem is well-known in dimensions n 2 5. (More detail and references appear in 8 1 below.) Perhaps the main result of this paper is that such manifolds are indeed locally flat provided they are freely embedded (i.e. that for each E > 0 there exists a map A : Mm x L”-m-’ --f (N” -Mm) such that h(x x Ln-m-l) is within E of x and links Mm homologically; here L”-m-’ is any simply connected manifold if n m 13 and is the sphere Sn-m-’ if n m 5 2); if n m = 2 we need the additional assumption that M”-* is locally flat at some point. A corollary is that freely embedded PL submanifolds are locally flat. The concepts of mapping cylinder neighborhood and freeness can be combined to obtain that of a strongly free submanifold Mm C N”: Instead of a mapping cylinder neighborhood, we require that there exist a map A of the mapping cylinder Z, into N” (where 7: Mm x L”-*-‘+ Mm is projection on the first factor) such that AIM” = identity and A(Z,, -Mm) C (N” -Mm), and A(6’(x)) links Mm. Another of our results is that strongly free submanifolds are locally flat (n # 4, with the flat spot proviso when n m = 2). These results each have local form. The method of proof in each case is to deduce that M” is locally homotopically unknotted in N”. In the course of the proof, criteria for commutativity of T,( U”), and others for the vanishing of 7~q( U”), 2 s q Ik, are developed for open orientable manifolds U” ; these criteria are perhaps of independent interest. For commutativity of T,( U”) we require that for any compact connected set C in U” there exist a compact connected orientable manifold K” with abelian fundamental group and a degree one map (K”, 8K”) --) (U”, U” C) which induces an injection H,(K”) + H1( U”). For vanishing of T~( U”)(2 5 q 5 k) we require that for any C there exist a compact connected orientable K” with TV = 0 (25 q 5 k) and a degree one map (K”, 8K”) + (U”, U” C) which induces an isomorphism T,(K”) + P,( U”). Conventions. Our notation is that of [23] with the. following exceptions. In all cases our homology is taken with integer coefficients and the coefficient group is suppressed from the notation. Our notation for the mapping cylinder Z.+ of a map 4: X + Y is as follows: Z, is the quotient space of XX [O, l] U Y X 2 in which points (x, 1) and (4(x), 2) are identified; X is identified with the image of X x 0 in Z+ ; and Y is identified with the image of Y x 2 in Z,. We use Z, to denote the image of X x [0, t] in 5. If M is a manifold we denote by aM the set of boundary points of M and set g = M JM.