- #1
Bacle
- 662
- 1
Hi, All:
I am trying to understand why covering maps have the
local triviality condition, i.e., given a cover C:X-->Y, every point y in Y
has a neighborhood Oy of y with p^-1(Oy)~ Oy x F, where F is the fiber. This
seems confusing, in that fibers of covering maps are a (discrete) collection of points
in X (since local diffeomorphisms are local bijections, the preimages are isolated/discrete)
. Does this statement just mean that the fiber is just a disjoint/discrete
collection of open sets, indexed by the cardinality of the fiber, or is there more than
that to it? I am thinking of standard examples like the covering map f:R-->$S^1$,
given by f(t)=(cost,sint), an infinite-to-one cover; would the Oy here be any 'hood
(neighborhood) of y that evenly-covers y?
Also: if this local triviality holds: is every covering map a bundle
of C over X with singletons as fibers?
Sorry, I know this is simple, but I haven't seen it in a while, and hopefully someone's
comments will jog my memory.
I am trying to understand why covering maps have the
local triviality condition, i.e., given a cover C:X-->Y, every point y in Y
has a neighborhood Oy of y with p^-1(Oy)~ Oy x F, where F is the fiber. This
seems confusing, in that fibers of covering maps are a (discrete) collection of points
in X (since local diffeomorphisms are local bijections, the preimages are isolated/discrete)
. Does this statement just mean that the fiber is just a disjoint/discrete
collection of open sets, indexed by the cardinality of the fiber, or is there more than
that to it? I am thinking of standard examples like the covering map f:R-->$S^1$,
given by f(t)=(cost,sint), an infinite-to-one cover; would the Oy here be any 'hood
(neighborhood) of y that evenly-covers y?
Also: if this local triviality holds: is every covering map a bundle
of C over X with singletons as fibers?
Sorry, I know this is simple, but I haven't seen it in a while, and hopefully someone's
comments will jog my memory.