Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma\_f\pi \colon \Sigma\_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma\_f X = X \times [0,1] / {\sim}$, where $\sim$ identifies $(x,0)$ with $(x',0)$ whenever $\pi(x) = \pi(x')$ and also $(x,1)$ with $(x',1)$ whenever $\pi(x) = \pi(x')$.


**Question**: Must $\Sigma\_f\pi \colon \Sigma\_f X \rightarrow Y$ also be a Serre fibration?


I think this is probably false for Hurewicz fibrations, and I would also be interested in counterexamples in that more general context.



Is the fiberwise suspension of a Serre fibration a Serre fibration?
Lemma 6 in Section 3 of


*Vandembroucq, Lucile*, [**Fibrewise suspension and Lusternik-Schnirelmann category**](https://doi.org/10.1016/S0040-9383(02)00007-1), Topology 41, No. 6, 1239-1258 (2002). [ZBL1009.55002](https://zbmath.org/?q=an:1009.55002).


(also available at <https://core.ac.uk/download/pdf/82041491.pdf>) seems to show that the fibrewise suspension of a Hurewicz fibration is a Hurewicz fibration. The proof uses lifting functions.


