Simplicial Complexes and Spaces
نویسندگان
چکیده
The notion of simplicial complexes is usually introduced at the graduate level, as part of the study of algebraic topology. So the main objective of this paper is to organize and present materials found in the literature so that it is accessible to students at the undergraduate level. We first generalize linear algebra to affine linear algebra. Then using these results, we define, and subsequently study, simplices. From there, we define simplicial complexes and provide a rigorous exposition of it. As an application, we introduce homology groups in the final chapter. Then in the last section of this paper, we illustrate in great detail how the homology groups of the real projective plane RP 2 can be computed using incidence matrices. The prerequisite for this paper is kept to the minimum, however it is assumed that the reader has already completed introductory courses in Real Analysis, Algebra and Linear Algebra. AFFINE LINEAR ALGEBRA One of the main limitations of linear subspace is that geometrically, it always passes through the origin. So we are not able to specify positions using the notion of linear algebra. This problem can be remedied by generalizing what we know about linear algebra to affine linear algebra. And fortunately many results in Affine Linear Algebra are analogous to those in Linear Algebra. In particular, we can define affine combinations, affine subspaces, affine bases and affine linear maps. As far as possible, we will extend results in Linear Algebra to prove the results in this chapter, so that the reader will be able to see the strong connections between them. Towards the end of this chapter, we also prove the continuity of linear maps and affine linear maps using the ε − δ definition, an approach which is not used by any of the literatures which we have consulted. SIMPLICIAL COMPLEXES Geometrically, simplices are triangles and tetrahedrons; algebraically, we can define simplices as the set 〈x0, ..., xq〉 := { q ∑ i=0 tixi ∈ R | ti ∈ R such that q ∑ i=0 ti = 1 and ti ≥ 0}, (1) where {x0, ..., xq} ⊂ R is an affinely independent set. The advantage of having an algebraic Student Associate Professor
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