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In mathematics, especially in homological algebra and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s ''X'' and ''Y'' and their product space X \times Y. In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer. A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic. These many results are named for the German mathematician Hermann Künneth.


Singular homology with coefficients in a field

Let ''X'' and ''Y'' be two topological spaces. In general one uses singular homology; but if ''X'' and ''Y'' happen to be
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es, then this can be replaced by
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
, because that is isomorphic to singular homology. The simplest case is when the coefficient ring for homology is a field ''F''. In this situation, the Künneth theorem (for singular homology) states that for any integer ''k'', :\bigoplus_ H_i(X; F) \otimes H_j(Y; F) \cong H_k(X \times Y; F). Furthermore, the isomorphism is a natural isomorphism. The map from the sum to the homology group of the product is called the ''cross product''. More precisely, there is a cross product operation by which an ''i''-cycle on ''X'' and a ''j''-cycle on ''Y'' can be combined to create an (i+j)-cycle on X \times Y; so that there is an explicit linear mapping defined from the direct sum to H_k(X \times Y). A consequence of this result is that the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s, the dimensions of the homology with \Q coefficients, of X \times Y can be determined from those of ''X'' and ''Y''. If p_Z(t) is the
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
of the sequence of Betti numbers b_k(Z) of a space ''Z'', then : p_(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 \infty, this reads as an identity on Poincaré polynomials. In the general case these are formal power series with possibly infinite coefficients, and have to be interpreted accordingly. Furthermore, the above statement holds not only for the Betti numbers but also for the generating functions of the dimensions of the homology over any field. (If the integer homology is not torsion-free, then these numbers may differ from the standard Betti numbers.)


Singular homology with coefficients in a principal ideal domain

The above formula is simple because vector spaces over a field have very restricted behavior. As the coefficient ring becomes more general, the relationship becomes more complicated. The next simplest case is the case when the coefficient ring is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princi ...
. This case is particularly important because the integers \Z are a PID. In this case the equation above is no longer always true. A correction factor appears to account for the possibility of torsion phenomena. This correction factor is expressed in terms of the Tor functor, the first derived functor of the tensor product. When ''R'' is a PID, then the correct statement of the Künneth theorem is that for any topological spaces ''X'' and ''Y'' there are natural short exact sequences :0 \to \bigoplus_ H_i(X; R) \otimes_R H_j(Y; R) \to H_k(X \times Y; R) \to \bigoplus_ \mathrm_1^R(H_i(X; R), H_j(Y; R)) \to 0. Furthermore, these sequences
split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, enterta ...
, but not canonically.


Example

The short exact sequences just described can easily be used to compute the homology groups with integer coefficients of the product \mathbb^2 \times \mathbb^2 of two
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
s, in other words, H_k(\mathbb^2 \times \mathbb^2; \Z). These spaces are
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es. Denoting the homology group H_i(\mathbb^2;\Z) by h_i for brevity's sake, one knows from a simple calculation with
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
that :h_0\cong \Z, :h_1\cong \Z/2\Z, :h_i= 0 for all other values of ''i''. The only non-zero Tor group (torsion product) which can be formed from these values of h_i is :\mathrm^_1(h_1, h_1) \cong \mathrm^_1(\Z/2\Z,\Z/2\Z)\cong \Z/2\Z. Therefore, the Künneth short exact sequence reduces in every degree to an isomorphism, because there is a zero group in each case on either the left or the right side in the sequence. The result is :\begin H_0 \left (\mathbb^2 \times \mathbb^2;\Z \right )\; &\cong \;h_0 \otimes h_0 \;\cong \;\Z \\ H_1 \left (\mathbb^2 \times \mathbb^2;\Z \right )\; &\cong \; h_0 \otimes h_1 \; \oplus \; h_1 \otimes h_0 \;\cong \;\Z/2\Z\oplus \Z/2\Z \\ H_2 \left (\mathbb^2 \times \mathbb^2;\Z \right )\; &\cong \;h_1 \otimes h_1 \;\cong \;\Z/2\Z \\ H_3 \left (\mathbb^2 \times \mathbb^2;\Z \right )\; &\cong \;\mathrm^_1(h_1,h_1) \;\cong \;\Z/2\Z \\ \end and all the other homology groups are zero.


The Künneth spectral sequence

For a general commutative ring ''R'', the homology of ''X'' and ''Y'' is related to the homology of their product by a Künneth spectral sequence :E_^2 = \bigoplus_ \mathrm^R_p(H_(X; R), H_(Y; R)) \Rightarrow H_(X \times Y; R). In the cases described above, this spectral sequence collapses to give an isomorphism or a short exact sequence.


Relation with homological algebra, and idea of proof

The chain complex of the space ''X'' × ''Y'' is related to the chain complexes of ''X'' and ''Y'' by a natural
quasi-isomorphism In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms :H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bul ...
:C_*(X \times Y) \cong C_*(X) \otimes C_*(Y). For singular chains this is the theorem of Eilenberg and Zilber. For cellular chains on CW complexes, it is a straightforward isomorphism. Then the homology of the tensor product on the right is given by the spectral Künneth formula of homological algebra. The freeness of the chain modules means that in this geometric case it is not necessary to use any hyperhomology or total derived tensor product. There are analogues of the above statements for singular cohomology and sheaf cohomology. For sheaf cohomology on an algebraic variety, Alexander Grothendieck found six spectral sequences relating the possible hyperhomology groups of two chain complexes of sheaves and the hyperhomology groups of their tensor product.


Künneth theorems in generalized homology and cohomology theories

There are many generalized (or "extraordinary") homology and cohomology theories for topological spaces. K-theory and
cobordism In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same d ...
are the best-known. Unlike ordinary homology and cohomology, they typically cannot be defined using chain complexes. Thus Künneth theorems can not be obtained by the above methods of homological algebra. Nevertheless, Künneth theorems in just the same form have been proved in very many cases by various other methods. The first were Michael Atiyah's Künneth theorem for complex K-theory and
Pierre Conner Pierre Euclide Conner (27 June 1932, Houston, Texas – 3 February 2018, New Orleans, Louisiana) was an American mathematician, who worked on algebraic topology and differential topology (especially cobordism theory). In 1955 Conner received his ...
and Edwin E. Floyd's result in cobordism. A general method of proof emerged, based upon a homotopical theory of modules over highly structured ring spectra. The homotopy category of such modules closely resembles the
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proc ...
in homological algebra.


References


External links

* {{DEFAULTSORT:Kunneth Theorem Homological algebra Theorems in algebraic topology