Computational Topology 6.1 Chains and Cycles
نویسنده
چکیده
In Lecture 3, we learned about a combinatorial method for representing spaces. In Lecture 4, we studied groups and equivalence relations implied by their normal subgroups. In this lecture, we look at a combinatorial and computable functor called homology that gives us a finite description of the topology of a space. Homology groups may be regarded as an algebraization of the first layer of geometry in cell structures: how cells of dimension n attach to cells of dimension n− 1 [1]. Mathematically, the homology groups have a less transparent definition than the fundamental group, and require a lot of machinery to be set up before any calculations. We focus on a weaker form of homology, simplicial homology, that both satisfies our need for a combinatorial functor, and obviates the need for this machinery. Simplicial homology is defined only for simplicial complexes, the spaces we are interested in. Like the Euler characteristic, however, homology is an invariant of the underlying space of the complex. Indeed, the invariance of the Euler characteristic is often derived from the invariance of homology. Homology groups, unlike the fundamental group, are abelian. In fact, the first homology group is precisely the abelianization of the fundamental group. We pay a price for the generality and computability of homology groups: homology has less differentiating power than homotopy. Once again, however, homology respects homotopy classes, and therefore, classes of homeomorphic spaces.
منابع مشابه
Topology Optimization of the Thickness Profile of Bimorph Piezoelectric Energy Harvesting Devices
Due to developments in additive manufacturing, the production of piezoelectric materials with complex geometries is becoming viable and enabling the manufacturing of thicker harvesters. Therefore, in this study a piezoelectric harvesting device is modelled as a bimorph cantilever beam with a series connection and an intermediate metallic substrate using the plain strain hypothesis. On the other...
متن کاملA New Topology for Z-Source Inverter Based on Switched-Inductor and Boost Z-Source Inverter
In this paper, a new topology for boost Z-source inverterbased on switched-inductor cell is proposed. The operating modes of the proposed inverter are analyzed and also a suitable control method to generate the trigger signals of the inverter is presented. Having a common earth between the input source and inverter and capability to generate a higher voltage gain by using lower amounts of the d...
متن کاملEffective Computational Geometry for Curves and Surfaces Chapter 7 Computational Topology: An Introduction
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 ...
متن کاملIsogeometric Topology Optimization of Continuum Structures using an Evolutionary Algorithm
Topology optimization has been an interesting area of research in recent years. The main focus of this paper is to use an evolutionary swarm intelligence algorithm to perform Isogeometric Topology optimization of continuum structures. A two-dimensional plate is analyzed statically and the nodal displacements are calculated. The nodal displacements using Isogeometric analysis are found to be ...
متن کاملTopology-Hiding Computation Beyond Logarithmic Diameter
A distributed computation in which nodes are connected by a partial communication graph is called topology-hiding if it does not reveal information about the graph (beyond what is revealed by the output of the function). Previous results [Moran, Orlov, Richelson; TCC’15] have shown that topology-hiding computation protocols exist for graphs of logarithmic diameter (in the number of nodes), but ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007