3 problems

Let $n-m\geq 3$. Write $\mathrm{V}^{pl}_{n,m}=\mathrm{PL}_n/\mathrm{PL}_{n,m}$ for the PL Stiefel space, and let $\mathrm{Operad}^h(E_m,E_n)$ be the derived mapping space of the li…

Let $\operatorname{Emb}(S^1,S^1\times S^3;S^1_0)$ be the based embedding space, and let $\operatorname{Diff}(S^1\times B^3\operatorname{fix}\partial)$ and…

Let the space of light bulbs be the space of properly embedded discs in $S^2\times D^2$ that agree with $x_0\times D^2$ near their boundary, with the appropriate topology. Non-simp…