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Properties: Non-orientable, has a single side. Fundamental group: $\pi_1(P^2) = \mathbb{Z}_2$

Properties: Non-orientable, no boundary. Fundamental group: $\pi_1(K) = \langle a, b | abab^{-1} = 1 \rangle$

Properties: Simplest closed and simply connected surface. Fundamental group: $\pi_1(S^2) = 0$ (trivial)

Properties: Cylinder-like surface, formed by two concentric circles in a plane. Fundamental group: $\pi_1(A) = \mathbb{Z}$

Properties: Infinite surface with constant negative curvature. Fundamental group: Depends on the specific tessellation or geometry.

Properties: Doughnut-shaped, genus 1 surface. Fundamental group: $\pi_1(T) = \mathbb{Z} \times \mathbb{Z}$

Properties: Surface with boundary, formed by extruding a circle along a line. Fundamental group: $\pi_1(C) = \mathbb{Z}$