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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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2 Answers 2

up vote 9 down vote accepted

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
3  
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
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But at the end where it says "Prop 1"it should say "Lemma 2". –  Tom Goodwillie Feb 1 '11 at 2:05

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