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If h preserves the path components then (M, c) is an n-dimensional manifold band, with the infinite cyclic cover M = F × R homotopy equivalent to a finite CW complex. Stallings [153] used codimension 1 surgery on a surface c−1 (∗) ⊂ M to prove that a map c : M −→S 1 from a compact irreducible 3-dimensional manifold M with ker(c∗ : π1 (M )−→Z) ∼ = Z2 is homotopic to the projection of a fibre bundle if and only if ker(c∗ ) is finitely generated, in which case ker(c∗ ) = π1 (F ) is the fundamental group of the fibre F .

Mather [91]. A manifold end of dimension ≥ 6 is tame if and only if × S 1 can be collared – this was already proved by Siebenmann [140], but the wrapping up procedure of Theorem 19 actually gives a canonical collaring of × S 1 . In principle, Theorem 19 could be proved using the canonical regular neighbourhood theory of Siebenmann [148] and Siebenmann, Guillou and H¨ahl [149]. We prefer to give a more elementary approach, using a combination of the geometric, homotopy theoretic and algebraic methods which have been developed in the last 25 years to deal with non-compact spaces.

The second proof outlined in [166, Remarks p. 189] uses more directly the existence of appropriate tubular neighbourhoods of strata called teardrop neighbourhoods. These neighbourhoods were shown to exist in the case of two strata by Hughes, Taylor, Weinberger and Williams [76] and in general by Hughes [74]. 13 we give a complete proof of the existence of teardrop neighbourhoods in the special case of the topologically stratified space (W ∞ , {∞}) determined by an open manifold W with a tame end.

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