Background:
In computer science, there is a notion of cyclic lists, which are typically implemented using pointers or modular arithmetic. Mathematically, we may define the set of 2-cycles of elements from a set $A$ as the quotient of $A\times A$ by the relations $(a_0,a_1)\sim(a_1,a_0)$. Similarly we may define the set of 3-cycles as the quotient of $A\times A\times A$ by the relations $(a_0,a_1,a_2)\sim(a_1,a_2,a_0)$. In homotopy type theory, the idea is to view all data types as "topological spaces", and it is possible to define the (pointed) circle $S^1$ as a data type.
Question:
In ordinary topology, let $S^1$ be the circle.
Define the quotient $C_2 :=S^1\times S^1/\sim$ where the equivalence relation $\sim$ is generated by $$(a_0,a_1)\sim(a_1,a_0)\qquad(a_0,a_1\in S^1).$$ What surface is it?
Similarly for $C_3 := S^1\times S^1\times S^1/\sim$ where the equivalence relation $\sim$ is generated by $$(a_0,a_1,a_2)\sim(a_1,a_2,a_0)\qquad(a_0,a_1,a_2\in S^1);$$ Can we identify this 3-dimensional space?