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For a more explicit example than Chris's, consider the map from the (2-dimensional) torus to a sphere that collapses the 1-skeleton of the usual CW complex and takes the 2-cell to the 2-cell of the sp …

There are lots of ways to do this; without more constraints (or indication of what is desired), it's hard to pick a best one. But here's one. We'll exploit the fact that for any two triangles, there …

There's another reference I'd like to promote: Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach, by Coufal, Pronk, Rovi, Scull, and Thatcher, in Women in topology: collaborati …

It's the same proof. Take a topological pants decomposition as before, and look for a minimal-length representative on your given Riemannian metric. Then you invoke the theorem that if you have a simp …

Any continuous map from M to N is homotopic to a smooth map, and if two smooth maps are homotopic, then they are also smoothly homotopic. (More generally, two homotopic functions are homotopic throug …

More useful than embedding it in some higher-dimensional space will be to give it as a manifold with identifications, I think. Topologically, $SO(3)/\!\sim$ is the same space as what you get by taking …

To expand on Michael Orlitzky's answer, unpacking the notation the claim is that, with $h_n(R)$ the $n\times n$ Jordan algebra of Hermitian matrices is $$ \begin{aligned} \mathrm{Aut}(h_n(\mathbb{R})) …