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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.

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I suspect that it's not the case. It's a theorem that there exists a functorial factorization into a trivial cofibration and a Serre fibration (which is what you want), but I don't know if one can explicitly compute it, since it results from the small-object argument. –  Harry Gindi Oct 2 '10 at 18:44
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2 Answers

up vote 9 down vote accepted

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
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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.

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I think I've managed to convince myself that this works by thinking about paths in $U$ that end on $C$. It would be nice if I could finish this proof with more precision though. –  Cary Oct 2 '10 at 21:45
    
Ah! Here's a potential problem: There are two possible topologies on the mapping cylinder of $E\to B$. The first one is the quotient topology of $E\times [0,1] \sqcup B$. In the second one, you declare any sequence of points whose t-coordinate tends to 1, and whose projection in B converges to converge. I think that my argument shows that the second topology on Mp is a Serre fibration... but I'm a bit confused by now. –  André Henriques Oct 3 '10 at 13:40
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You may want to look at Proposition 1.3 in: Mónica Clapp, Duality and transfer for parametrized spectra. Arch. Math. (Basel) 37 (1981), no. 5, 462--472. She proves that a suitable pushout of Hurewicz fibrations over a common base is a Hurewicz fibration. By a theorem of Steinberger and West, a Serre fibration between CW complexes is a Hurewicz fibration, so if $p \colon E \to B$ is a cellular map of CW complexes, and a Serre fibration, then these results prove that $M_p \to B$ is a Hurewicz fibration, hence a Serre fibration. –  John Rognes Oct 4 '10 at 9:09
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