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If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder of $p$ also a fibration?

I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The Homotopy Category is a Homotopy Category") that the topology of $M_p$ can be slightly modified to make $q$ a fibration.

I wonder

  1. whether the modification is necessary: does anyone know of a fibration $p$ such that $q$ is not a fibration, and

  2. whether the modification is necessary if we assume the spaces are compactly generated (Strøm works with all spaces).

EDIT: There is a natural injective map from mapping cylinder $M_f$ of the map $f: X\to Y$ to the join $X * Y$; the topology on $M_f$ is modified so that it coincides with the subspace topology.

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If your fibration $p:E\to B$ had fibre $F$ then $q:M_p \to B$ has pointwise fibre $CF$, the cone on $F$. If $p$ is locally trivial then $q^{-1}(U)\cong U\times CF$. I don't see where the modification of the topology of $M_p$ comes in this case at least! –  Somnath Basu Dec 15 '10 at 17:35
    
Yes, this was what I was trying to say: $q$ is a fibration if $p$ is locally trivial, without any modification. –  Jeff Strom Dec 15 '10 at 18:01
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If $E$, $B$ are metrizable and $p$ is uniformly continuous, and you replace quotient topology by the topology of quotient uniformity everywhere, then your two mapping cylinders will be homeomorphic (and also metrizable). The preprint about this is not on the arxiv yet (delayed by other sections) but I can share it. In fact, the two mapping cylinders are uniformly homeomorphic. So is there a nice model category structure on metrizable uniform spaces and u.c. maps? Unfortunately, not all pushouts are metrizable, though all adjunction spaces are. I wonder if some trick can save pushouts. –  Sergey Melikhov Dec 17 '10 at 2:07
    
@Sergey: this sounds very interesting! I can wait for your paper to make it to the arXiv -- no rush for me. –  Jeff Strom Dec 17 '10 at 2:12
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The preprint mentioned in my previous comment is at arxiv.org/abs/1106.3249 –  Sergey Melikhov Sep 29 '11 at 15:03
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