4 problems

Let $\mathsf{CW}(2,3)$ be the homotopy category of one-connected three types, namely spaces $X$ with $\pi_iX=0$ for all $i\not=2,3$. Let $\mathsf{SG}$ be the category of square gro…

Let $G(n)$ be the quotient of the free group $F(x_1,\ldots,x_n)$ by the relations that $x_1\cdots x_n=1$ and every word that becomes trivial when any one generator is replaced by t…

Let $SPT_*$ be the simplicial group of simplicial pure twin groups, let $c_{111}$ be the 2-simplex with trivial faces, and let $K_*$ be the smallest simplicial subgroup of $SPT_*$…

Classifying-map conjecture. Under the relevant zigzag of equivalences, the map $N\alpha_*$ is a classifying map for $p:E\to B$: its homotopy class coincides with…