query: Why is the fundamental group of the circle so non-trivial?
positive_passages.docid: 131519
positive_passages.text: intuition on the fundamental group of $S^1$
negative_passages.docid: 69652
negative_passages.text: Help understanding proof of every topological group is regular
negative_passages.docid: 149029
negative_passages.text: 3-manifold examples of homomorphisms between fundamental groups that are not induced by continuous maps
negative_passages.docid: 43659
negative_passages.text: Why is $\pi$ so close to $3$?
negative_passages.docid: 116759
negative_passages.text: is the group of rational numbers the fundamental group of some space?
negative_passages.docid: 135126
negative_passages.text: Proving that the quotient manifold is orientable if and only if the group action is orientation-preserving
negative_passages.docid: 11000
negative_passages.text: Fundamental group of $\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\})$
negative_passages.docid: 74978
negative_passages.text: How the center of a non-cyclic free Group is trivial?
negative_passages.docid: 37635
negative_passages.text: Diffeomorphism group of the unit circle
negative_passages.docid: 50587
negative_passages.text: Example of a domain in R^3, with trivial first homology but nontrivial fundamental group
negative_passages.docid: 52802
negative_passages.text: Why is the image of $k^+$ dense in the character group?
negative_passages.docid: 154990
negative_passages.text: Fundamental group of two tori with a circle ($S^1✕${$x_0$}) identified
negative_passages.docid: 142698
negative_passages.text: Proving that the fundamental groups of two spaces with same homotopy type are isomorphic
negative_passages.docid: 99689
negative_passages.text: Zeroth homotopy group: what exactly is it?
negative_passages.docid: 55652
negative_passages.text: Fundamental group of the torus
negative_passages.docid: 113919
negative_passages.text: Surface of genus $g$ does not retract to circle (Hatcher exercise)
negative_passages.docid: 94215
negative_passages.text: Fundamental group of the connected sum of manifolds