An Introduction to Algebraic Topology
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Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex.
Applied Algebraic Topology
For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes. Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space.
It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes.
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This result means that we can combine the theoretical power of singular homology and the computability of simplicial homology to get many applications. Description In this course we develop two important ways to study a topological space: 1 via its coverings and its fundamental group, 2 via its singular homology.
Hours of class per week 2 Final grade Homework and final exam oral or written, depending on the number of students. Prerequisites Basic theory of groups as in Algebra 1, basic theory of vector spaces as in Lineaire algebra 1 — 2, basic theory of topological spaces as in Topologie. Literature We will treat parts of J.
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Examples include the plane , the sphere , and the torus , which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Knot theory is the study of mathematical knots.
Homology Theory: An Introduction to Algebraic Topology
While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points , line segments , triangles , and their n -dimensional counterparts see illustration. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.
The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
An Introduction to Algebraic Topology - Joseph J Rotman - Bok () | Bokus
A CW complex is a type of topological space introduced by J. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes , but still retains a combinatorial nature that allows for computation often with a much smaller complex.
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An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones  the modern standard tool for such construction is the CW complex. In the s and s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups , which led to the change of name to algebraic topology.
In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism or more general homotopy of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statement easier to prove. Two major ways in which this can be done are through fundamental groups , or more generally homotopy theory , and through homology and cohomology groups.
The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with.
The fundamental group of a finite simplicial complex does have a finite presentation. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with. In general, all constructions of algebraic topology are functorial ; the notions of category , functor and natural transformation originated here.