Basic Algebraic Topology
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Nowadays that includes fields like physics, differential geometry, algebraic geometry, and number theory. As an example of this applicability, here is a simple topological proof that every nonconstant polynomial p z has a complex zero. Consider a circle of radius R and center at the origin of the complex plane. The polynomial transforms this into a closed curve in the complex plane. If this image curve ever passes through the origin, we have our zero.
Well, suppose the radius R is very large. Then the highest power of p z dominates and hence p z transforms the circle into a curve which winds around the origin the same number of times as the degree of p z.
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This is called the winding number of the curve around the origin. It is always an integer and it is defined for every closed curve which does not pass through the origin.
Algebraic Topology -- from Wolfram MathWorld
If we deform the curve, the winding number has to vary continuously but, since it is constrained to be an integer, it cannot change and must be a constant unless the curve is deformed through the origin. Now deform the image curve by shrinking the radius R to zero and suppose that the image curve never passes through the origin, that is to say that we never get the zero of the polynomial. The image curve gets very small, hence must have winding number 0 around the origin unless it is shrinking to the origin.
If the image curve is shrinking to the origin, the origin is a zero. If not, the winding number is 0 which means that the polynomial must have degree 0, in other words, it is a constant.
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The winding number of a curve illustrates two important principles of algebraic topology. First, it assigns to a geometric odject, the closed curve, a discrete invariant, the winding number which is an integer. Second, when we deform the geometric object, the winding number does not change, hence, it is called an invariant of deformation or, synomynously, an invariant of homotopy.
Modern algebraic topology is the study of the global properties of spaces by means of algebra. Poincare' was the first to link the study of spaces to the study of algebra by means of his fundamental group.
This is a generalization of the concept of winding number which applies to any space. To get an idea of what algebraic topology is about, think about the fact that we live on the surface of a sphere but locally this is difficult to distinguish from living on a flat plane. Clover calculus for homology 3-spheres via basic algebraic topology. Article information Source Algebr. Export citation. Export Cancel. URL: Link to item. Info Print Print.
Table Of Contents. Submit Feedback. Thank you for your feedback. Introduction Basic concepts of general topology Simply connected Topological equivalence Homeomorphism Topological structure Topological space Continuity Algebraic topology Fundamental group Differential topology Knot theory History of topology. Load Previous Page.
Algebraic topology The idea of associating algebraic objects or structures with topological spaces arose early in the history of topology. Load Next Page. More About.