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I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...

Let $X,Y$ be compact smooth manifolds. Let $f,g\colon X\to Y$ be smooth submersions (in particular, locally trivial bundles) which are homotopic to each other (in the class of smooth maps, not ...