Abe homotopy classification of topological excitations under the topological influence of vortices

Topological excitations are usually classified by the $n$th homotopy group $\pi_n$. However, for topological excitations that coexist with vortices, there are case in which an element of $\pi_n$ cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of $\pi_n$ corresponding to the charge of a topological excitation may change when the topological excitation circumnavigates a vortex. This phenomenon is referred to as the action of $\pi_1$ on $\pi_n$. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group $\kappa_n$. The $n$th Abe homotopy group $\kappa_n$ is defined as a semi-direct product of $\pi_1$ and $\pi_n$. In this framework, the action of $\pi_1$ on $\pi_n$ is understood as originating from noncommutativity between $\pi_1$ and $\pi_n$. We show that a physical charge of a topological excitation can be described in terms of the conjugacy class of the Abe homotopy group. Moreover, the Abe homotopy group naturally describes vortex-pair creation and annihilation processes, which also influence topological excitations. We calculate the influence of vortices on topological excitations for the case in which the order parameter manifold is $S^n/K$, where $S^n$ is an $n$-dimensional sphere and $K$ is a discrete subgroup of $SO(n+1)$. We show that the influence of vortices on a topological excitation exists only if $n$ is even and $K$ includes a nontrivial element of $O(n)/SO(n)$.Comment: 36 pages, 12 figure

Retroperitoneal lipoma arising from the urinary bladder

Retroperitoneal benign lipomas are extremely rare and represent about 2.9% of all primary retroperitoneal tumors. About 80% of the tumors in the retroperitoneal cavities are malignant neoplasms. We experienced a case of a retroperitoneal lipoma simulating an ovarian mature cystic teratoma. A diagnosis was correctly made by magnetic resonance imaging (MRI) prior to surgery, and a total tumorectomy was performed. The retroperitoneal lipoma was recognized to have arisen from the urinary bladder. Histological sections revealed a tumor consisting of typical adipose cells without atypia. These types of lipomas should be carefully followed-up because they often recur and undergo malignant transformations