Universal coefficient theorem
Encyclopedia
In mathematics
, the universal coefficient theorem in algebraic topology
establishes the relationship in homology theory
between the integral homology of a topological space
X, and its homology with coefficients in any abelian group
A. It states that the integral homology groups
completely determine the groups
Here
might be the simplicial homology
or more general singular homology
theory: the result itself is a pure piece of homological algebra
about chain complex
es of free abelian group
s. The form of the result is that other coefficients A may be used, at the cost of using a Tor functor
.
For example it is common to take A to be
, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion
in the homology. Quite generally, the result indicates the relationship that holds between the Betti number
s bi of X and the Betti numbers bi,F with coefficients in a field
F. These can differ, but only when the characteristic
of F is a prime number
p for which there is some p-torsion in the homology.
.
The theorem states that there is an injective group homomorphism
ι from this group to
, which has cokernel
.
In other words, there is a natural short exact sequence
Furthermore, this is a split sequence
(but the splitting is not natural).
The Tor group
on the right can be thought of as the obstruction to ι being an isomorphism.
, stating that there is a natural short exact sequence
As in the homological case, the sequence splits, though not naturally.
, the real projective space
. We compute the singular cohomology of X with coefficients in
.
knowing that the integer homology is given by:
We have
, so that the above exact sequences yield
.
In fact the total cohomology ring
structure is
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the universal coefficient theorem in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although algebraic topology...
establishes the relationship in homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
between the integral homology of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X, and its homology with coefficients in any abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
A. It states that the integral homology groups
completely determine the groups
Here
might be the simplicial homologySimplicial homology
In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory....
or more general singular homology
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
theory: the result itself is a pure piece of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
about chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
es of free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
s. The form of the result is that other coefficients A may be used, at the cost of using a Tor functor
Tor functor
In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....
.
For example it is common to take A to be
, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsionTorsion
The word torsion may refer to the following:*In geometry:** Torsion of a curve** Torsion tensor in differential geometry** The closely related concepts of Reidemeister torsion and analytic torsion ** Whitehead torsion*In algebra:** Torsion ** Tor functor* In medicine:** Ovarian...
in the homology. Quite generally, the result indicates the relationship that holds between the Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s bi of X and the Betti numbers bi,F with coefficients in a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F. These can differ, but only when the characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
of F is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p for which there is some p-torsion in the homology.
Statement
Consider the tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
.The theorem states that there is an injective group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
ι from this group to
, which has cokernelCokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
.In other words, there is a natural short exact sequence
Furthermore, this is a split sequence
Splitting lemma
In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....
(but the splitting is not natural).
The Tor group
Tor functor
In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....
on the right can be thought of as the obstruction to ι being an isomorphism.
Universal coefficient theorem for cohomology
There is also a universal coefficient theorem for cohomology involving the Ext functorExt functor
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
, stating that there is a natural short exact sequence
As in the homological case, the sequence splits, though not naturally.
Example: mod 2 cohomology of the real projective space
Let, the real projective spaceReal projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
. We compute the singular cohomology of X with coefficients in
.knowing that the integer homology is given by:
We have
, so that the above exact sequences yield.In fact the total cohomology ring
Cohomology ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
structure is
.