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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 ...

Let $S_1 = S_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p_i : S_1 \times S_2 \to S_i$ be the projection, $j : U \to S_1 \times S_2$ the inclusion of the complement of the diagonal and $...