$p:E \rightarrow B$ is a fibration and $F$ is its fibre and $F_p$ its homotopy fibre. If $i:F \rightarrow F_p$ is the inclusion, is there a homotopy inverse $r$ of $i$ such that $r \circ i = id$?
Fibre and homotopy fibre
3
$\begingroup$
algebraic-topology
homotopy-theory
fibration
-
0What happens in the case where your fibration is a trivial vector bundle over a connected base space? Take the cylinder for example. What happens here? – 2012-08-30
-
0See Hatcher, Proposition 4.65 – 2012-08-30
-
0@JuanS Hatcher's book proves that $F$ is homotopy equivalence to $F_p$ by proving fibre homotopy equivalence without the last restriction. – 2012-08-30
-
0Maybe i'm being slow, but doesn't a homotopy equivalence give you exactly what you want? – 2012-08-31
-
0I think in some cases it does. For example when $(F_p,F)$ satisfies homotopy extension property, $F$ is the deformation retract of $F_p$. – 2012-08-31