Magical Triangles
نویسنده
چکیده
Simplicial complexes consist of a set of vertices together with designated subsets. They can be thought of as embedded in Rn with the induced metric and topology, where n is large enough that the points can all be geometrically independent. The sets of points spanned by these designated subsets then form triangles or their higher or lower dimensional analogues according to some restrictions. By gluing together simplices in various ways, many compact manifolds can be approximated up to homeomorphism by finite complexes. In addition, we show that any simplicial complex of dimension n can be realized in R2n+1 without compromising the basic structure of the complex, regardless of number of vertices. Because their component parts are fairly simple, approximation with simplices can make it easier to compute properties of a space, such as the Euler characteristic. Continuous maps between spaces can also be approximated up to homotopy by linear simplicial maps, which map the simplicial structure of one space into another.
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