# Quail Entry: Deck and Cov. Sp. Action

this happened december 17th

### Deck Transformations

rep. by commutative triangles (only one covering space involved!)

a deck transformation is a lift of the projection .

via. lifting property if is path conn. then determined by a point in the sense that if agree on a point then they are equal.

normal means essentially transitive, any two points in a fiber can be mapped to each other via a deck transfomration.

A covering space is normal, ‘normal’ is not applied to deck transformations.

Reason for ‘normal’ : a normal covering space iff is normal subgroup in .

hypotheses: to prove need lifting criterion which requires path connected and locally path conn. (but by general properties of path conn. cov. space I guess loc. path. is necessarily inherited by ) so require loc. path connected and restrict to path connected cov. spaces to apply result.

regard pf: let be points in the fiber of want to know when there is comm. triangle with corners by lift crit this happens when in . Upstairs these are related by conjugating by a path ; downstairs this is conjugation by a loop.

is image of downstairs. If , then above implies deck tranf. taking a pt in fiber to where is a lift starting at $y$. In fact for gp of deck tranf

And isomorphism:

pf: uses image of are just loops downstairs that lift to loops upstairs.

### Covering Space Actions

require: acts on and has nbd s.t. iff .

In particular the collection are disjoint and becomes a covering space.

1. Its always a normal covering space.

path conn. implies deck transf. det. by image of point

2.

To use iso need to be path. conn and loc. path conn to get

3.

where is as above and using normality here.

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- Published:
- January 1, 2010 / 8:11 pm

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- Quail, wall scribble

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