Algebraic Topology: Oaxtepec 1991 by Tangora M.C. (ed.)

By Tangora M.C. (ed.)

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With this in mind, let us make more precise the definition of a map, which so far we have thought of as a union of properly attached polygons. We begin with the standard n-gons Sn , which may be thought of, for instance, as the regular n-gons lying in the complex plane C with vertices at the nth roots of unity exp(2πik/n), 1 ≤ k ≤ n. As was mentioned before, we allow the case n = 2, which may be thought of as the unit circle {z ∈ C : |z| = 1} with two vertices at ±1 and two edges, one the top half of the circle, the other the bottom half; we also allow the case n = 1, which may be thought of as having the entire unit circle as its single edge, and z = 1 as its single vertex.

In particular, since the map (x, y) → (2x, 2y) is a covering map from the flat torus to itself, we have 4χ(T2 ) = χ(T2 ), which provides an alternative proof that χ(T2 ) = 0. 3. Lecture 10: Wednesday, Sept. 19 a. From triangulations to maps. Given two surfaces M1 and M2 equipped with triangulations T1 and T2 , if f : M1 → M2 is a homeomorphism, then f (T1 ) is a triangulation of M2 , hence χ(T1 ) = χ(T2 ). It follows that χ(M1 ) = χ(M2 ), so Euler characteristic is a topological invariant of a compact triangulable surface.

2) Any point x ∈ M which is not a vertex has at most two preimages; in particular, it lies in at most two of the Pi . The latter condition ensures that each edge is identified with at most (in fact, exactly) one other. With the precise definition in hand, we can now state the following: Theorem 2. Let M be a compact surface which admits a triangulation T , and let M be any map on M . Then χ(M) = χ(T ), and hence any two such maps have the same Euler characteristic. Proof: Proceed exactly as in the proof of Theorem 1, with T1 = T and T2 = M, noting that we may just as easily approximate the map M with the mesh T1n as the triangulation T2 .

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