Fundamental groups of the configuration spaces of all triangles and right triangles
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This is a question from a past comprehensive exam: Consider triangles in the plane, with vertices given by non-colinear points as usual. The space $T$ of all plane triangles can be given a natural quotient topology: $T$ is the quotient of an open subset of $mathbb{R}^6=(mathbb{R}^2)^3$ by the action of the symmetric group $S_3$ permuting the vertices of a triangle. Let $R$ be the space of all right triangles in the plane. Let $i:Rto T$ be the inclusion map. Is the map $i_*:pi_1(R)to pi_1(T)$ surjective? I do not know the answer and I only have a very vague picture of $T$ and $R$ as some fiber bundles. I don't really know where to start when faced with such problems. Any hint will be appreciated.
algebraic-topology fundamental-groups...
