7 problems

Kearnes' conjecture. Every F-quasigroup is isotopic to a Moufang loop.

Let $1.13b$ denote the smooth approximation theorem discussed in the source: in the range $m>\frac{3(n+1)}2$, smoothly $\delta$-isotopic smooth embeddings are smoothly…

Let $\Sigma\subseteq S^4$ be a locally flat, embedded, closed, orientable surface of genus $g$, and suppose that $\pi_1(S^4\setminus\Sigma)\cong\mathbb{Z}$ is generated by a meridi…

Let $B_n(S^2)$ be the braid group of $n$ points on the sphere, with $n\geq 3$, and let $\Delta^2\in B_n(S^2)$ denote its non-trivial central element. Write $\widehat{\Delta^2}$ for…

Let $p>3$ be a prime, and consider right Bol loops of order $3p$, with two such loops identified when they are isotopic. Enumeration claim. The number of right Bol loops of order…

Let $M$ be a non-orientable surface, and let $\gamma$ be a contractible simple closed curve on $M$. Baer-type conjecture. Theorem $1.1$ is true for $\gamma$, namely, the isotopy co…

Nonexistence conjecture. For an arbitrary $n$ with $n\geq n_0$, where $n_0$ is sufficiently large, the isotopy of Theorem 1 is not valid, even without the additional requirements t…