Re: Coordinate charts for torus

grubb@xxxxxxxxxxxxxxxxx (Daniel Grubb) writes:


>>>>What is the minimum number of coordinate charts needed to cover the doughnut
>>>>surface T^2? Could you also explain why, informally? Thank you.
>>> I don't see a reason you'd need more than one.
>>> A computer screen which "wraps" both top-to-bottom and left-to-right maps to
>>> a torus.
>
>
>>Because each coordinate patch ought to be mapped by a diffeomorphism
>>to a chunk of the torus. Using a single 'patch' is going to be
>>many-one in places.
>
>If you think iof the torus as the square with opposite sides identified,
>then the interior of the square gives a coordinate patch that covers
>everything except for two intersecting circles. A translate of this patch
>will similarly cover all except for two circles. The two will not cover
>all except two points, which a third patch can easily cover. Hence
>three patches will do.

And in fact three patches (each homeomorphic to R^2) will do for
any connected 2-manifold, and more genrally n+1 for any connected
n-manifold (smooth or piecewise linear, anyway; and let's assume
compact for safety's sake).

>I will assume that patches are homeomorphic to R^2, and so are
>homotopically trivial. Since the 'cup length' of the torus is 3,
>i.e. cup products of three cohomology classes are 0, but there
>is a cup product of two that is non-zero, an MV argument shows that
>you need at least 3 patches to cover.

It's good I read this thread to the end before jumping in, as it
saves me the effort of introducing "cup length" myself. I will
only add that, by dropping DG's assumption "that patches are
homeomorphic to R^2", one can find a set of two patches that
cover the torus, each homeomorphic to an open annulus. One
of the definitions of "coordinate patch" merely requires it
to be an open subset of R^n for the appropriate n, so I'm
not being too radical here.

Lee Rudolph
.

Relevant Pages

  • Re: Coordinate charts for torus
    ... >to a chunk of the torus. ... Using a single 'patch' is going to be ... three patches will do. ... i.e. cup products of three cohomology classes are 0, ...
    (sci.math)
  • Re: Coordinate charts for torus
    ... Since the 'cup length' of the torus is 3, ... >i.e. cup products of three cohomology classes are 0, ... >you need at least 3 patches to cover. ... Prev by Date: ...
    (sci.math)
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