Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology.

Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory. A manifold is a topological space that near each point resembles Euclidean space.

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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.

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.

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 [2] 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.

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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.

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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.

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## Algebraic Topology (Master)

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. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence or, much more deeply, existence of mappings.

One of the first mathematicians to work with different types of cohomology was Georges de Rham. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the s, when Samuel Eilenberg and Norman Steenrod generalized this approach.

## Basic Algebraic Topology

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms e. MIT faculty and instructors have gone on to make connections with still more elaborate and contemporary segments of arithmetic algebraic geometry, and are now in the process of reworking this entire area, creating a deep unification of algebraic geometry and algebraic topology. This work will form the foundation of much research over the next decade, and offers the promise of providing tools useful in algebra as well as in topology.

Specifically, our group works in stable and unstable homotopy theory, homotopical group theory, higher category theory, derived algebraic geometry, elliptic cohomology, computational homotopy theory and string topology. Research and training in Geometry and Topology, from the undergraduate to the postdoctoral level, are supported by an RTG grant from the National Science Foundation. Massachusetts Institute of Technology Department of Mathematics. Faculty Haynes Miller.