Three-sheeted connected covering spaces of the wedge of three real projective spaces

The title is pretty self-explanatory: the exercise is to classify such coverings.

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Three-sheeted connected covering spaces of the wedge of three real projective spaces

One thought on “Three-sheeted connected covering spaces of the wedge of three real projective spaces

  1. fmartin says:

    Some thoughts:

    By the Galois correspondence between connected covering spaces and subgroups of the fundamental group, the original problem is equivalent to finding all subgroups of index 3 of the fundamental group of the space. By Van Kampen’s theorem, the fundamental group of the wedge of three real projective spaces is isomorphic to the free product of three copies of \mathbb{Z}/2\mathbb{Z}, which we will denote G.

    The problem is then reduced to finding all subgroups of G of index 3. Now, any normal subgroup with this property induces an epimorphism G\to \mathbb{Z}/3\mathbb{Z}. However, since the free product is the coproduct in the category of groups, any map G\to\mathbb{Z}/3\mathbb{Z} is induced by three maps \mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/3\mathbb{Z} and no non-trivial maps exist between those two groups, so \mathrm{Hom}(G,\mathbb{Z}/3\mathbb{Z})=0. This proves that no normal three-sheeted coverings exist, but there may be non-normal subgroups of index 3 of G.

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