# Quasifibrations and homotopy pullbacks

I'm wondering about the theoretical placement of quasifibrations.

One nice thing about "weak fibrations" (maps homotopy equivalent in the category of maps to Hurewicz fibrations) is that a pullback square involving (one) weak fibration is a homotopy pullback square.

Is the corresponding result true for quasifibrations in the Serre-Quillen context? That is, suppose $E\to B$ is a quasifibration, and the square $$\begin{array}{ccc} P & \to & E \cr\downarrow&pb&\downarrow \cr X& \to &B \end{array}$$ is a categorical pullback. Then is it a homotopy pullback in the Quillen-Serre model structure?

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The definition of quasifibration (according to Dold & Thom, 1958) is: a map $f:E\to B$ such that for all $b$ in $B$, the canonical map from the fiber to the homotopy fiber is a weak equivalence. Pullbacks with respect to such maps are not generally homotopy pullbacks; an example was given in that 1958 paper (Bermerkung 2.3), which goes something like this:

Let $\newcommand{\R}{\mathbb{R}}B=\R\times \R$. Then $E$ will have the same underlying set as $B$, and $f$ will be the identity map. But we topologize $E$ by "tearing" along the positive $y$-axis. For instance, let $E$ have the smallest topology such that $f$ is continuous and the set $[0,\infty)\times (0,\infty)$ is open.

The space $E$ is still contractible with this topology (it deformation retracts to $\R\times -1$). Therefore, the homotopy fiber over any point of $B$ must be weakly contractible, and thus weakly equivalent to the actual one-point fiber. So $f$ is a quasi-fibration.

Let $X= \mathbb{R}\times 1\subset B$, and let $P$ be the pullback of $E$ over $X$. Then $P$ has two path components, while $X$ is contractible; this is not a homotopy pullback!

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This answers my question, and makes me wish I could read German much more quickly. – Jeff Strom Jan 30 '11 at 22:50

It rather depends what you mean by quasi-fibration. The most useful resource I know for questions about quasi-fibrations is this message of Goodwillie, posted to the APGTOP mailing list in 2001.

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This is nice to have in my back pocket, thanks. – Jeff Strom Jan 30 '11 at 22:50
Wow, I enjoyed writing that, but I had lost track of it. It's nice to see it again. – Tom Goodwillie Jan 31 '11 at 1:58
But at the end where it says "Prop 1"it should say "Lemma 2". – Tom Goodwillie Feb 1 '11 at 2:05