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I am studying a problem in hyperplane arrangement theory related to the homotopy type of the complement manifold of a certain class of hyperplane arrangements.

In a well celebrated paper Richard Randell proved that, given a smooth one-parameter family of arrangements $\mathcal{A}_{t}$ in $\mathbb{C}^{d}$ with constant lattice of intersection, then for any time $t_{1}$ and $t_{2}$ the complement manifolds $M(\mathcal{A}_{t_{1}})$ and $M(\mathcal{A}_{t_{2}})$ are diffeomorphic. For more references and more precise definitions and notations the original article is available at this link.

The smoothness hypothesis on the one-parameter family $\mathcal{A}_{t}$ is deeply exploited. To be more precise, the proof uses Thom's first isotopy theorem and Whitney stratification in the context of smooth manifolds.

$\textbf{Question:}$ Is that possible to relax the smoothness hypothesis on the one-parameter family $\mathcal{A}_{t}$ obtaininig a weaker result?

In particular, I hope that something similar to the following $\textbf{desired statement}$ holds. However, I am looking for confirmations and bibliographical references.

$\textbf{Desired statement:}$ Given a continuous one-parameter family of arrangements $\mathcal{A}_{t}$ in $\mathbb{C}^{d}$ with constant lattice of intersection, then for any time $t_{1}$ and $t_{2}$ the complement manifolds $M(\mathcal{A}_{t_{1}})$ and $M(\mathcal{A}_{t_{2}})$ have the same homotopy type.

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  • $\begingroup$ You can try to write down a map whose fibers are these complements and try to show that it's, say, a Serre fibration. The nice thing about the smooth case is that you can appeal to tools like Ehresmann's fibration theorem to do this. $\endgroup$ May 11 '15 at 17:31

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