By Henryk Torunczyk, J. West

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**Example text**

Then by Fibred Stability (B7), there is a fiber-preserving homeomorphism h:E -» E x Q. Now the natural map fine fiber homotopy equivalence. 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.

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.

J c-» X x U U * EIT, and note that we may assume that connected open set of Q. j) V be a neighborhood of intersects and N N x {b} X. b € W. As W E, of a in E|W <^-> N x W ^ Proposi tion. fibration, let A (1) Given W Since E|U V; here dN E|W Let A r|N x W p:E -» B b X. X x U in a is the boundary of E, C N x {0} N for Ln X. to X is paracompact, and Then there is an extension X Hence, be a locally compact ANR- be a map, where f € Map R (A,E) o£ is closed in r x id w . a w is the desired diagram. p':X -* B U in further if necessary we easily get a be a closed set in neighborhood such that is connected this implies Restricting let U E, fl 6N x {b} = (|) for fiber-preserving homotopy from (A10) in b € V.