In topology, the dunce cap is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Simply gluing two sides oriented in the same direction would yield a cone much like the layman's dunce cap, but the gluing of the third side results in identifying the base of the cap with a line joining the base to the point. For the fundamental group, use the fact that we can find a homotopy between X and the wedge of S2 and S1 (by moving the points where the chord joins S2 so they coincide). The fundamental group of the wedge S2 and S1 is the free product of π1 (S1) and π1 (S2) = 1, which is π1 (S1) = Z. In this paper we prove that the fundamental group of the n-fold dunce cap is a cyclic group of order n and the fundamental group of the torus is a free abelian group of rank 2.
