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.



Is the mapping cylinder of a Serre fibration also a Serre fibration?
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


