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at Serre fibration I have spelled out the proof that that with $F_x \hookrightarrow X \overset{fib}{\to} Y$ then $\pi_\bullet(F) \to \pi_\bullet(X)\to \pi_{\bullet(Y)}$ is exact in the middle. here.
(This is intentionally the low-technology proof using nothing but the definition. )
Added to Serre fibration a section Properties – Relation to Hurewicz fibrations with pointer to
a) counter-example of a Serre fibration that is not a Hurewicz fibration
b) statement that all Serre fibrations between CW-complexes are Hurewicz fibrations.
Also added to the Examples-section pointer to the homotopy lifting property for covering spaces.
(The projection map of) a fibre bundle over any paracompact space, or more generally, a bundle that admits a trivialisation over a numerable cover, is a Hurewicz fibration, I believe.. In general, all fibre bundles are Serre fibrations.
(I was torn between adding this comment here or at the covering space thread)
Fixed some syntax errors
Removed the references to May’s Concise course, since it does not treat Serre fibrations (cf. §7.1: “Serre fibrations are more appropriate for many purposes, but we shall make no use of them.”); recognition over numerable convers is a theorem about Hurewicz fibrations, so it would fit better on that page. (For a map $f\colon X\to Y$ to be a Serre fibration it suffices for $f|_{f^{-1}(U)}f^{-1}(U)\to U$ to be a Serre fibration for all $U$ in any open cover of $Y$; to sketch the construction of the lift: subdivide the $I^{n+1}$ into a grid of closed subcubes whose image lies in some $U$, and construct the lift inductively, starting in the corner with the origin.)
Anonymous
Thanks for the alert. I’ll fix it when I am back online (on my phone now).
Put in a reference for the statement I just gave to tom Dieck’s 2008 book. The stuff about numerable covers is now at Hurewicz fibration.
(Hope I got the formatting right – I’m a long time reader, first time editor…)
Anonymous
Thanks a million! For all this (catching it, fixing it across pages, and last not least, for setting me straight – much appreciated).
I have expanded out the statement at local recognition a little, just for beautification.
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