# Surfaces

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## Abstract

We now turn our attention to a more macroscopic view of spaces, and try to find more global and geometric descriptions of the objects of study. It is still our purpose to try to find topological invariants of the shape of a space, but the determination of whether two objects are topologically equivalent is an often difficult, indeed in many cases as yet unsolved, problem. In this text we will use what is called the combinatorial approach to topology, thus limiting ourselves to a large but relatively manageable group of objects. All the spaces we study will be built from a uniform set of building blocks. We then analyze the blocks themselves and how they are combined. The basic building blocks are called cells and are assembled into complexes. The cells are of varying dimensions, and so a complex has a stratified structure: a natural ordering of the cells by dimension. This text concentrates on spaces which are locally 2-dimensional.

### Keywords

• Topological Space
• Projective Plane
• Simplicial Complex
• Cell Complex
• Edge Point

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“...did anyone know, he enquired, what a cubic foot of hardwood weighed? While all the women were respectfully waiting for the men to answer this, and all the men were knitting their brows, and trying to look as if they really did know, but had forgotten for the moment, Biddy suddenly said in a bewildered voice that the [tree] trunk was round and she thought cubes were square? Bruce replied quite civilly that cubes were cubic, and Biddy retorted that anyway, they weren’t round, and she didn’t see what square cubes had to do with a round trunk. Bruce, in a tone of slightly strained patience, pointed out that the trunk was not round, but cylindrical, and Biddy said that even if it was, all the cylinders she had ever seen were round, so what were they arguing about? This led to a general and extremely animated discussion (accompanied by much drawing of diagrams on the ground, and much quoting of such examples as pennies, wedding-rings, tennis balls, and water pipes)...from which it seemed to transpire, rather to everyone’s astonishment, that simply nothing at all was round.”

Eleanor Dark

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### Cite this chapter

Kinsey, L.C. (1993). Surfaces. In: Topology of Surfaces. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0899-0_4