I am doing an exercise in a representation theory book that asks the following: "For $g$ and $h$ in $SL_{2}\mathbb{C}$, the mapping $A \mapsto gAh^{-1}$ is in $SO_{4}\mathbb{C}$. Show that this gives a 2:1 covering $SL_{2}\mathbb{C} \times SL_{2}\mathbb{C} \rightarrow SO_{4}\mathbb{C}$." What does "2:1 covering" mean? How does one show such a thing?
What does "2:1 covering" mean?
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general-topology
representation-theory
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4"$2:1$" means the preimage of every point consists of two points. "covering" means it is a covering map: for every point in the $q$ image, there is a neighborhood $V$ of $q$ such that the inverse image of $V$ is a disjoint union of open sets, on each of which the map induces a homeomorphism onto $V$. – 2012-03-04
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0Double-wrapped. – 2012-03-04
1 Answers
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"2:1" means the inverse image of each point is a set of two points. "Covering" means it's surjective and probably enjoys other nice properties like continuity and being locally one-to-one. Take a look at this article: http://en.wikipedia.org/wiki/Covering_space