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Effective Computational Geometry for Curves and Surfaces Chapter 7 Computational Topology : An Introduction
| Content Provider | Semantic Scholar |
|---|---|
| Author | Rote, Günter Vegter, Gert |
| Copyright Year | 2007 |
| Abstract | approach is followed in the context of singular homology theory. This theory is more powerful when proving general results like topological invariance of homology spaces. Since we focus on basic computational techniques we will not discuss this theory here, but refer the reader to standard textbooks on algebraic topology, like [11]. The equivalence of Simplicial and Singular Homology is proven in [11, Sect. 2.1]. Chain spaces and simplicial homology. Let K be a finite simplicial complex. In this chapter, an simplicial k-chain is a formal sum of the form ∑ j ajσj over the oriented k-simplices σj in K, with coefficients aj in the field Q of rational numbers. In other words, it can be regarded as a rational vector whose entries are indexed by the oriented k-simplices of K. Furthermore, by definition, −σ = (−1)σ is the simplex obtained from σ by reversing its orientation. With the obvious definition for addition and multiplication by scalars (i.e., rational numbers), the set of all simplicial k-chains forms a vector space Ck(K,Q), called the vector space of simplicial k-chains of K. The dimension of this vector space is equal to the number of k-simplices of K. Therefore, the Euler characteristic of a d-dimensional simplicial complex K can be expressed as an alternating sum of dimensions of the spaces of k-chains: |
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| Alternate Webpage(s) | http://page.mi.fu-berlin.de/rote/Papers/pdf/Computational+topology:+an+introduction.pdf |
| Alternate Webpage(s) | http://www.inf.fu-berlin.de/~rote/Papers/pdf-gzipped/Computational+topology:+an+introduction.pdf.gz |
| Alternate Webpage(s) | http://www.inf.fu-berlin.de/~rote/Papers/pdf/Computational+topology:+an+introduction.pdf |
| Alternate Webpage(s) | http://page.inf.fu-berlin.de/~rote/Papers/pdf/Computational+topology:+an+introduction.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Chapter |
