Narrow Results

We develop a theory on a topologically nontrivial manifold which leads to different vacuum backgrounds at the field level. The different colors of the same quark flavor live in different backgrounds generated by the action of the torsion subgroup of H²(M, 2πℤ) on H²(M, 2πℤ) itself. This topological separation leads to a quark confinement mechanism which does not apply to the baryons as they turn out to live on the same vacuum state. The theory makes some topological assumptions on the spacetime manifold which are compared with the available data on the topology of the Universe.

It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an $m$-almost quasigroup ($m$-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.

It is known that the order of a finite quandle annihilates its reduced quandle homology and the torsion subgroup of its rack homology if the quandle is quasigroup. However, this does not hold in general if a quandle is connected. In this paper, we prove that under a certain condition, the reduced quandle homology and the torsion subgroup of the rack homology of a connected quandle are annihilated by the order of the inner automorphism group of the quandle.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $G$ be the torsion group $E{(K)}_{\mathrm{\text{tors}}}$ for some cubic field $K$ which does not occur over $\mathbb{Q}$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\mathbb{Q}$ together with cubic fields $K$ so that $G=E{(K)}_{\mathrm{\text{tors}}}$.