6 problems

Let $\Omega_n=X({\tt K}_n)$ be the cycle-free chessboard complex, and let … Here, a space is $\mu_n$-connected when its homotopy groups vanish through dimension $\mu_n$. Best-conne…

Freeness conjecture. The reduced homology group $\widetilde H_i(M_{m,n})$ is free if and only if $i\geq m-2$.

Boundary-range conjecture. There is some $m_0$ such that, if $n_0=2m_0-6$ or $n_0=2m_0-7$, then $\widetilde H_{\nu_{m_0,n_0}}(M_{m_0,n_0})$ is an elementary $3$-group.

Exponent-three conjecture. The exponent in Theorem $$ is $3$.

Let … -complex, with vertex set the pinched braids having precisely one regular strand. Sphericality conjecture. … is large enough. The paper presents this as evidence toward under…

Sharp connectivity conjecture. The connectivity bound given in Theorem is the best possible; equivalently, for each $n\geq 2$,