By Abhay Ashtekar

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Fiber-preserving homeomorphism trivialization of 8. p x id Q :E x Q -» B x Q is a By Proposition (B6) there is a g:E x Q -» B x Q. ) A GENERAL POSITION FIBRATION THAT IS NOT A BUNDLE. In this section we give an example of a General Position Fibration p:E -* B over a one-dimensional Peano continuum (the "Hawaiian Earring") with connected, one-ended Q-manifold that is not a locally trivial bundle. fibers We construct it by stringing together mapping tori of homeomorphisms of a compact Q-manifold that are homotopic but sometimes not isotopic to the identity.

If 44 H. TORUNCZYK AND J. WEST f:E~ -* E- is any proper fine fiber homotopy equivalence, then 0 for every open cover homeomorphism n: of_ E.. there is a fiber-preserving ^ n -> E- fl-close to f. We now obtain from this fibred homeomorphism theory that, just as for locally trivial bundles, pull-backs of General Position Fibrations by homotopic maps are homeomorphic by fiber preserving homeomorphisms. 5) Theorem. Let p:E -* B be a proper locally compact ANR- fibration satisfying Fibred General Position.

ANR- fibration satisfying Fibred General Position. *ExIxI-*ExI is stationary on a subset taken stationary on (3) extends f~ be Then o_f_ E x I p'-fiber-preserving ambient isotopv Xft, X 1 Then for each open there is a fiber-preserving (over f:E x I -* E B) 0-close to the projection. 1), and Bing's Shrinking Criterion for locally compact ANR-fibrations (A8), we may follow the proof given in [E] (cf. [Tl]). 1), obtaining Fibred Z-set Unknotting, with control, in General Position Fibrations. 6) and Bing's Shrinking Criterion (A8) in this context.

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