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# 数学代写|微分拓扑作业代写differential topology代考|Compact Surfaces

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## 数学代写|微分拓扑作业代写differential topology代考|The Klein Bottle

To simplify we could imagine that we were two-dimensional beings living in a static closed surface. The sphere and the torus are familiar surfaces, but there are many more. If you did Example 1.4.2, you were exposed to another surface, namely the Klein bottle. This has a plane representation very similar to that of the torus: just reverse the orientation of a single edge Figure 1.14.

Although the Klein bottle is an easy surface to describe (but frustrating to play chess on), it is too complicated to fit inside our three-dimensional space again a manifold is not a space inside a flat space, it is a locally Euclidean space. The best we can do is to give an “immersed” i.e., allowing self-intersections picture Figure 1.15.

Speaking of pictures: the Klein bottle makes a surprising entré in image analysis. When analyzing the nine-dimensional space of $3 \times 3$ patches of gray-scale pixels, it is of importance – for instance if you want to implement some compression technique – to know what high-contrast configurations occur most commonly. Carlsson, Ishkhanov, de Silva and Zomorodian show in [5] that the subspace of “most common high-contrast pixel configurations” actually “is” a Klein bottle.

Their results have been used to develop a compression algorithm based on a “Klein bottle dictionary”.

## 数学代写|微分拓扑作业代写differential topology代考|Classification of Compact Surfaces

As a matter of fact, it turns out that we can write down a list of all compact surfaces compact is defined in Appendix A, but informally should be thought of as “closed and of bounded size”. First of all, surfaces may be divided into those that are orientable and those that are not. Orientable means that there are no loops by which two-dimensional beings living in the surface can travel and return home as their mirror images. (Is the universe non-orientable? Is that why some people are lefthanded?

All connected compact orientable surfaces can be obtained by attaching a finite number of handles to a sphere. The number of handles attached is referred to as the genus of the surface.

## 数学代写|微分拓扑作业代写DIFFERENTIAL TOPOLOGY代考|Plane Models

If you find such descriptions elusive, you may derive some comfort from the fact that all compact surfaces can be described similarly to the way we described the torus. If we cut a hole in the torus we get a handle. This may be represented by plane models as in Figure 1.19: identify the edges as indicated.

If you want more handles you just glue many of these together, so that a $g$-holed torus can be represented by a $4 g$-gon where two and two edges are identified.

It is important to have in mind that the points on the edges in the plane models are in no way special: if we change our point of view slightly we can get them to be in the interior.

## Matlab代写

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