Is the mapping cylinder of a Serre fibration also a Serre fibration? If we have a Serre fibration $p: E \rightarrow B$ with fiber of homotopy type $S^{k-1}$, then we can create a fibration with contractible fiber by first taking the mapping cylinder $M_p$ of $p$ to get a map $M_p \rightarrow B$ (not necessarily a fibration) with contractible "fiber" and then applying the "space of paths" construction to get a fibration $M_p \times_B B^I \rightarrow B$. My question is, is this last part of the construction necessary, or is the mapping cylinder $M_p$ already a Serre fibration?
I tried lifting a homotopy $f_t: X \times I \rightarrow B$ with starting point $\tilde f_0: X \rightarrow M_p$ by cutting $X$ into the closed preimage $C$ of $B \subset M_p$ and the open preimage $U$ of $E \times [0,1) \subset M_p$. On $C \times I$ we set $\tilde f_t(x) = f_t(x) \in B \subset M_p$. On $U \times I$ we lift $f_t|U: U \times I \rightarrow B$ to $g_t: U \times I \rightarrow E$ and then set $\tilde f_t(x) = (g_t(x),(1-t)(t$ coordinate of $\tilde f_0(x)) + t(1))$. This defines a continuous lift on $C$ and on $U$ separately. If the continuous lift on $U$ extends to the closure of $U$ then we're done. The map $U \rightarrow E$ could be nasty though near the boundary of $U$. Perhaps a better approach is to first construct a map from $X \times I$ that is only "close to" a lift, then use obstruction theory (I'm not an expert on this) to show that it is homotopic to some lift.
 A: Waldhausen, Jahren and myself proved a
fiber gluing lemma for Serre fibrations,
in the context of simplicial sets, that
may be useful.  In Propositions 2.7.10
and 2.7.12 of
"Spaces of PL manifolds and categories
of simple maps"
http://folk.uio.no/rognes/papers/plmf.pdf
we prove that given:


*

*a diagram of
simplicial sets $Z_1 \leftarrowtail
Z_0 \to Z_2$, where one map is a
cofibration,

*a sufficiently
nice base simplicial set $B$ (a
simplicial complex will do), and

*compatible maps $Z_i \to B$ that become
Serre fibrations upon geometric
realization,
then the pushout map
$Z_1 \cup_{Z_0} Z_2 \to B$ becomes
a Serre fibration upon geometric
realization.
Mapping cylinders are a special
case of pushouts.  If $p \colon E \to B$
becomes a Serre fibration upon
realization, then so do
the obvious map $E \times \Delta^1
\to B$ and the identity map
$B \to B$.  The pushout map is
your map $M_p \to B$, and our
conclusion is that its realization
is a Serre fibration.
Our proof depends on working with
simplicial sets.  The technical
condition on $B$ is that each
nondegenerate simplex is embedded.


*

*John

A: I think that your idea works.
You propose "$\tilde f_t(x) = (g_t(x),(1-t)(t$ coordinate of $\tilde f_0(x)) + t(1))$", which I don't quite understand. I would simply do:
$\tilde f_t(x) = (g_t(x),t$ coordinate of $\tilde f_0(x))$,
where $g_t(x)$ is the solution of the corresponding lifting problem for $p$.
Note that you've implicitely used that $U$ is a CW-complex, but there's nothing wrong with that.
As you say, this defines a continuous lift on $C$ and on $U$ separately, and they glue to a continuous lift on the whole.
