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Silviu

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In summary: The cohomology of a manifold is determined by the cohomology of the n-simplexes. - The tensor product of two manifolds is antisymmetric. However, the tensor product of r-forms is not antisymmetric. - The tensor product of two r-forms is antisymmetric. However, the tensor product of two n-forms is not antisymmetric. - The tensor product of two r-forms is antisymmetric. However, the tensor product of two n-forms is symmetric.

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Silviu

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mathwonk

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Silviu said:

If ##π_{M}## is the projection map of ##M×N## onto ##M## then ##π_{M}^{*}## maps ##H^{*}(M)## into ##H^{*}(M×N)##. Similary ##π_{N}^{*}## maps ##H^{*}(N)## into ##H^{*}(M×N)##.

These pullback maps determine a bilinear mapping of ##H^{*}(M)×H^{*}(N)→H^{*}(M×N)## by ##([α],[β])→[(π_{M}^{*}α)∧π_{N}^{*}β]##. Writing this in terms of the grading of cohomology dimensions, one has maps ##Σ_{i+j=k}H^{i}(M)⊗H^{j}(N)→H^{k}(M×N)## for each dimension ##k##.

BTW: The Kunneth Theorem applies to any Cartesian product of manifolds not just to 3 manifolds that are Cartesian products..

- The cohomology determined by differential forms is called De Rham cohomology. It is a theory that is defined only for differentiable manifolds. Singular cohomology is another cohomology theory. It is defined for all topological spaces. De Rham cohomology is isomorphic to singular cohomology with real numbers as coefficients. The idea of the proof is to view differential forms as homomorphisms of the groups of smooth n- simplexes into the real numbers - or into the complex numbers.

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The Kunneth formula is a mathematical tool used in algebraic topology to study the cohomology of spaces that are products of other spaces. It relates the cohomology of the product space to the cohomology of its factors. In particular, it provides a way to compute the cohomology of a product space using the cohomology of its factors.

The tensor product in r-forms is a mathematical operation that combines two r-forms to create a new r-form. It is defined as a bilinear map that takes two r-forms as inputs and produces a new r-form as an output. In the context of the Kunneth formula, the tensor product is used to compute the cohomology of product spaces.

The Kunneth formula involves taking the tensor product of cohomology classes in the cohomology rings of each factor space. This allows for the computation of the cohomology of the product space in terms of the cohomology of its factors. In other words, the Kunneth formula uses the tensor product operation to relate the cohomology of a product space to the cohomology of its factors.

The Kunneth formula is an important tool in algebraic topology, as it provides a way to compute the cohomology of a product space using the cohomology of its factors. This has numerous applications in various areas of mathematics, including differential geometry, algebraic geometry, and algebraic number theory.

Yes, the Kunneth formula has been generalized to other mathematical structures, such as sheaf cohomology, etale cohomology, and de Rham cohomology. These generalizations allow for the computation of cohomology of more complex spaces, and have important applications in fields such as algebraic number theory and algebraic geometry.

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