Consider a locally trivial (topological) bundle over the Klein bottle $$ I\to E \to K$$ The projection map $E \to K$ is a homotopy equivalence. Is it a simple homotopy equivalence? Due to the dimension of the base (2), one cannot use Whitehead torsion to check simple homotopy equivalence.
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2$\begingroup$ Triangulate the bundle so that projection is simplicial. Inducting downwards, use that there is a way to push in simplices on $\Delta \times I$ to deformation retract this onto $\Delta \times {1/2} \cup \partial \Delta \times I$, where the {1/2} should be the 0 section. $\endgroup$– mmeCommented Nov 9, 2018 at 22:15
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