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• lichen1983312
In summary, cohomology theory is a practical and important tool in mathematics and physics. It allows for the study of spaces and their topology by using abstract constructions such as cochain groups and homomorphisms. The cup product in cohomology provides more information than homology and can be used to understand the topology of a space. There is no general procedure for finding the cochain group or cohomology group of a space, but techniques like Mayer-Vietoris sequences can be used to break down a space into simpler pieces and then combine the results. In certain cases, like the torus, intersection counts can be used to map homology classes to integers or mod 2 values.
lichen1983312
I am self leaning some basic cohomology theory and I managed to go through from the definition to the universal coefficient theorem. But I don't think I get the main point of this theory, I like to ask this questions:
Is such an abstract theory practical?

I would say that homology is practical, because the chain groups could be built on the basis of maps from simplexes to the space, which is intuitive and easy to operate. Since the boundary operator also has clear geometric meaning, at least in theory one can just write down the chain group and follow a well defined procedure to compute the homology group.

For the cochain groups ##C_n^ * = Hom({C_n},G)## , the elements are just homomorphisms. This construction is too abstract that I cannot see what is this group look like and how is cohomology gorup ##\ker /im## computed. If one cannot easily do these things, why would we even need this theory ?

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Cohomology groups have a natural multiplication called the cup product that turns them into a graded algebra. These products give more information about a space than the homology groups. There are even examples of spaces with identical homology (with any coefficients) but whose cup product structure is not the same. This is one reason why cohomology is important.

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For oriented compact manifolds, integer cohomology classes can be expressed as intersections of cycles of complementary dimension. If you think of trying to completely separate two submanifolds of complementary dimension then the intersection number (the oriented count of the number of intersection points) is an obstruction to being able to do this. If you ignore the orientation then you get a cohomology class with ##Z_2## coefficients. This tells you something about the topology of the manifold.

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lavinia said:
For oriented compact manifolds, integer cohomology classes can be expressed as intersections of cycles of complementary dimension. If you think of trying to completely separate two submanifolds of complementary dimension then the intersection number (the oriented count of the number of intersection points) is an obstruction to being able to do this. If you ignore the orientation then you get a cohomology class with ##Z_2## coefficients. This tells you something about the topology of the manifold.
Thanks very much, I need time to understand all this. But before that, given a space, is there a general procedure to find the co-chain group or cohomology group?

lichen1983312 said:
Thanks very much, I need time to understand all this. But before that, given a space, is there a general procedure to find the co-chain group or cohomology group?

Not that I have seen. One technique that often works is to divide the space up into subspaces whose cohomology is easy and then fit the pieces together. This can often be done with a Mayer-Vietoris sequence. This sequence can be used for both homology and cohomology.

lichen1983312 said:
Thanks very much, I need time to understand all this. But before that, given a space, is there a general procedure to find the co-chain group or cohomology group?
A simple case is the torus. Its first homology group is generated by two orthogonal circles that intersect in a single point. These circles can be taken to be the equator of the torus and the orthogonal circle that cycles through the hole. Intersecting homology classes with either one of the circles and counting the number of intersection points (with orientation) produces a homomorphsim of the first homology group into the integers.

The homomorphism corresponding to one of the circles maps the homology class of that circle to zero and the homology class of the orthogonal circle to 1. The interesting thing to see is that the intersection count is always the same for any representative of either homology class as long as the intersection is transversal - that is: as long as the two curves cross over each other at each intersection point.

For a homomorphisminto into ##Z_{2}## assign 1 if the number of intersections is odd and 0 if the number of intersections is even. Note for instance that if one distorts one of the two orthogonal circles so that it intersects the other in more than one point, an even number of new intersection points is created so the mod 2 count is unchanged. The oriented count is also unchanged.

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1. What is cohomology?

Cohomology is a mathematical concept used in topology and algebraic geometry. It is a way to measure the "holes" in a space, or the number of independent cycles that exist in the space.

2. How is cohomology different from homology?

Cohomology is the dual concept to homology. While homology measures the "holes" in a space, cohomology measures the "filled-in" regions of a space. In other words, cohomology is the study of the "co-cycles" of a space, which are the boundaries of the cycles measured by homology.

3. What are the applications of cohomology?

Cohomology has many applications in mathematics, physics, and engineering. It is used to solve problems in topology, algebraic geometry, differential equations, and more. In physics, cohomology is used to study symmetries and conservation laws in physical systems.

4. How is cohomology calculated?

Cohomology is calculated using a mathematical tool called a cochain complex. This complex consists of a sequence of vector spaces and linear maps, which are used to determine the cohomology groups of a given space.

5. What is the importance of cohomology in mathematics?

Cohomology is an important concept in mathematics because it allows us to understand the global structure of a space by studying its local properties. It also provides a powerful tool for solving problems in various branches of mathematics and has many applications in other fields.

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