What Is the Use of a Space Covering?

[quasar987]

What is the use of having a covering of a space?

 

[matt grime]

How else do you calculate the fundamental group?

 

[quasar987]

How you do extract the fact that the fundamental group of S^1 is Z from the fact that it is covered by R (naturally).

In class, we have only had time to coverer the basics of the basics, i.e. the definition, the unique path lifting property and the homotopy lifting property. And the final's question on coverings was simply, "Find a 5-sheeted covering of S^1 x S^1 by S^1 x S^1" haha.

 

[matt grime]

The group of deck transformations (or the homotopy lifting property stuff). It isn't that the cover is R, but that the covering fibres have certain properties. You can use this to work out the fundamental groups of S^1, tori, genus n surfaces, klein bottles, mobius bands etc.

 

[quasar987]

Mmmh, ok but all those are extremely easily obtained with Van Kampen. I'm sure there is more to it than just another (and more complicated it seems!) way to find the fundamental group.

 

[matt grime]

And how did you find the fundamental groups of the 'simpler' objects? Please post a proof that the fundamental group of S^1 is Z using the van kampen theorem. Anyway, it is much of a muchness, and one man's easy is another man's hard. To me it is clear from the covering space and its deck transformations what the fundamental groups of various spaces (that aren't all amenable to van kampen) are. Van kampen requires that you can find subspaces whose intersection is connected, and whose fundamental groups you can identify. That is not always possible.

 

[quasar987]

Ok, fair enough.

 

[matt grime]

There is of course more to it than topology: an elliptic curve is, amongst other things, a 2:1 cover of P^1 branched at 4 points. Covers then let you work out the genus of curves, for example. This is the Riemann Hurwitz theorem. (More of a mathwonk area than mine.)

 

[mathwonk]

well, of course the map from R to S^1, taking t to (cost,sint) is familiar to very unsophisticated students who have never seen van kampens theorem .

 

[mathwonk]

covers also let you define the fundamental group in abstract algebraic geometry and number theory where covers exist but "loops" do not.

 

[Hurkyl]

Local homeomorphisms are rather nice functions to study. Surjective maps are also rather nice. And since covers are both...

For some lowbrow points, covers aren't just good for computing with, but they're good for computing in. For example, we can more easily manipulate angular displacement when working in the cover Z-->S^1. And covers provide a natural setting for studying the behavior of multivalued functions like the complex logarithm. I've found that computing in the cover R^2-->RxS^1 to be very useful in working through the cosmological twin paradox.

 

[mathwonk]

coverings allow one to use geometric reasoning to be applied to algebra. for instance the standard proof that a free group on 2 generators contains a subgroups isomorphic to a free group on any finite number n of generators uses the fact that a wedge of 2 circles has a cover by a space homotopic to a wedge of n circles.

 

[mathwonk]

the standard tool of polar coordinates in analytic geometry, is nothing but a covering space of the punctured plane by the full plane.

it represents difficult shapes like circles by easy shapes like line segments.

 

[mathwonk]

here is an interesting theorem in complex analysis, whose proof can be summanrized quickly using covering spaces.

riemann showed there are only three simply connected complex surfaces, the complex numbers C, the unit disc D, and the complex sphere (projective space) P.

Moreover every connected complex surface admits a covering by exactly one of these.

We know an entire function can omit two values on the complex sphere, since e^z does so. The reason for this is that C is the simply connected surface that covers P minus 2 points.

But no non constant entire function can omit three values in P, since the covering of P minus three points, is given by D. Thus if an entire function f omits three points, then it factors through a map to D, which must be constant.


Riemann method of studying study complex surfaces was to cover them by simpler surfaces. In the case of compact connected surfaces, the only one covered by P is P, and the only ones covered by C are elliptic curves. The ones covered by D are all Riemann surfaces of genus more than 1.

The different covers lead to a study of the different actions on D by discrete groups of automorphisms, which is a branch of non euclidean geometry.

the gauss bonnet theorem for instance shows that the only surfaces that can be covered by the disc have genus at least 2, since the poincare metric on D descends to the covered space, which must then have negative euler characteristic.

covering space theory is probably the most useful tool in all of global complex analysis.

here for example is a proof of the fundamental theorem of algebra using it. A complex polynomial defines a holomorphic map from P to P, which is a covering space away fromj the images of the critical points. since a covering space has the same number of preimages of every point, the map is thus either onto or constant.

 

[mathwonk]

here is another proof of FTA using global branchged covering methods: a non constant polynomial defiens a map from P to P which is both open and closed, hence surjective, since P is connected.

 

[Chris Hillman]

Suggest two good books

What is the use of having a covering of a space?

Just thought I'd suggest two good books which answer this question very nicely:

Algebraic topology, an introduction, by William S. Massey, Springer, 1989

(warning--- only covers homotopy theory, but does this very well indeed)

Algebraic topology, by Allen Hatcher, Cambridge University Press, 2002