Under what conditions for a space $M$ does the projection map to the first factor $p: M \times M - \Delta \rightarrow M$ has the local triviality condition, i.e. is a fiber bundle? Where $\Delta$ denotes the diagonal $\{(a,a) \}_{a \in M}$.
Fiber bundle M x M - diagonal
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general-topology
fiber-bundles
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0Interesting question. Certainly looks true for closed manifolds $M$. – 2011-09-19
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1If $M$ is a closed manifold we can do the following: Let $U_i$ be a chart in $M$, so we can assume $U_i\cong \mathbb{R}^n$. Now consider the neighborhood $\mathbb{R}^n \times \mathbb{R}^n - \Delta \cong U_i \times U_i - \Delta \subset M \times M - \Delta$. Showing the result for $\mathbb{R}^n \times \mathbb{R}^n - \Delta$ and using a bump function inside an open set of $U_i \times U_i - \Delta$ we can can conclude that $p^{-1}(U_i)\cong \mathbb{R}^n \times (\mathbb{R}^n- \mathbb{R}^{n-1})$. – 2011-09-19
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0Oops, I meant $p^{-1}(U_i)\cong \mathbb{R}^n \times (\mathbb{R}^n-\{0\})$. And of course, we can ignore the $i$ index of the chart $U_i$... – 2011-09-19