Timeline for Is the mapping cylinder of a Serre fibration also a Serre fibration?
Current License: CC BY-SA 2.5
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| Oct 4, 2010 at 9:09 | comment | added | John Rognes | You may want to look at Proposition 1.3 in: Mónica Clapp, Duality and transfer for parametrized spectra. Arch. Math. (Basel) 37 (1981), no. 5, 462--472. She proves that a suitable pushout of Hurewicz fibrations over a common base is a Hurewicz fibration. By a theorem of Steinberger and West, a Serre fibration between CW complexes is a Hurewicz fibration, so if $p \colon E \to B$ is a cellular map of CW complexes, and a Serre fibration, then these results prove that $M_p \to B$ is a Hurewicz fibration, hence a Serre fibration. | |
| Oct 3, 2010 at 13:40 | comment | added | André Henriques | Ah! Here's a potential problem: There are two possible topologies on the mapping cylinder of $E\to B$. The first one is the quotient topology of $E\times [0,1] \sqcup B$. In the second one, you declare any sequence of points whose t-coordinate tends to 1, and whose projection in B converges to converge. I think that my argument shows that the second topology on Mp is a Serre fibration... but I'm a bit confused by now. | |
| Oct 2, 2010 at 21:45 | comment | added | Cary | I think I've managed to convince myself that this works by thinking about paths in $U$ that end on $C$. It would be nice if I could finish this proof with more precision though. | |
| Oct 2, 2010 at 18:40 | history | answered | André Henriques | CC BY-SA 2.5 |