In mathematics, the Künneth theorem of algebraic topology describes the singular homology of the cartesian product Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...

X × Y

of two topological spaces, in terms of singular homology groups H_i(X, R) and H_j(X, R). Here (not to strive for the greatest generality) R is a given commutative ring of coefficients. Even in the case where R is the ring Z of integers, the statement of the full result requires some use of homological algebra, namely use of the Tor functors. From now on the coefficients in R will always be tacitly understood in the notation. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... The Tor functors are the derived functors of the tensor product functor in mathematics. ...

If R is taken to be a field then there is no need to invoke the Tor functors. The result in this case can be used as a 'first approximation' to the general case. It states that In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...

H_k(X × Y)

is the direct sum of the tensor products of the vector spaces In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... The fundamental concept in linear algebra is that of a vector space or linear space. ...

H_i(X)

with

H_j(Y)

for all pairs (i,j) with

i + j = k.

Further there is a cross product operation showing how an i-cycle on X and a j-cycle on Y can be combined to create an (i + j)-cycle on X × Y; so that there is an explicit linear mapping defined from the direct sum to H_k(X × Y). The statement of the Künneth theorem, when R is a field, is that this linear mapping is an isomorphism. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

As a consequence the Betti numbers of X × Y are determined by those of X and of Y; the statement is formally equivalent to saying that if p_Z(t) is the generating function of the sequence of Betti numbers B_i of a space Z, then In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

p_{X × Y} (t) = p_X(t)p_y(t).

Here when there are finitely many Betti numbers of X and Y, each of which is a natural number rather than ∞, this reads as an identity on Poincaré polynomials. In the general case these are formal power series with possible coefficients ∞, and have to be interpreted accordingly. Furthermore, the Betti numbers over any field F, b_F, satisfy the same type of relationship as the usual b_Q for rational number coefficients (they need not actually be the same numbers unless the homology is torsion-free). Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

To extend this to the case of general R, it is necessary to change the statement: the R-module homomorphism defined φ in just the same way by the cross product is injective, and there is a description now of its cokernel. That is, we have to define an R-module T by the direct sum of the In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ...

Tor_R(H_p(X), H_q(Y))

taken over all the p and q with

p + q = k − 1.

Then the cokernel of φ is isomorphic to T. Therefore, in any case where the relevant Tor groups can be shown to vanish, we do have an isomorphism. This is not, however, universally true (in the early days of algebraic topology the phenomena caused by torsion in homology groups, of which this is one, appeared subtle and misled researchers). In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...

The usual proof of the result depends on the Eilenberg-Zilber theorem.

The result is named for the German mathematician Otto Hermann Künneth (1892-1975). The idea of a Künneth formula has now become a generic term, applied to many homological theories.