Adams operation
Encyclopedia
In mathematics
, an Adams operation
is a cohomology operation
in topological K-theory
, or any allied operation in algebraic K-theory
or other types of algebraic construction, defined on a pattern introduced by Frank Adams
. The basic idea is to implement some fundamental identities in symmetric function
theory, at the level of vector bundle
s or other representing object in more abstract theories. Here k ≥ 0 is a given integer.
The fundamental idea is that for a vector bundle V on a topological space X, we should have
as
of the roots α of a polynomial
P(t). (Cf. Newton's identities
.) Here Λk denotes the k-th exterior power. From classical algebra it is known that the power sums are certain integral polynomials Qk in the σk. The idea is to apply the same polynomials to the Λk(V), taking the place of σk. This calculation can be defined in a K-group, in which vector bundles may be formally combined by addition, subtraction and multiplication (tensor product
). The polynomials here are called Newton polynomials (not, however, the Newton polynomial
s of interpolation
theory).
Justification of the expected properties comes from the line bundle
case, where V is a Whitney sum of line bundles. For that case treating the line bundle direct factors formally as roots is something rather standard in algebraic topology (cf. the Leray–Hirsch theorem). In general a mechanism for reducing to that case comes from the splitting principle
for vector bundles.
Adams operations can be defined more generally in any λ-ring over the rational numbers.
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...
, an Adams operation
- ψk
is a cohomology operation
Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...
in topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
, or any allied operation in algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
or other types of algebraic construction, defined on a pattern introduced by Frank Adams
Frank Adams
John Frank Adams FRS was a British mathematician, one of the founders of homotopy theory.-Life:He was born in Woolwich, a suburb in south-east London. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his Ph.D. from the University of...
. The basic idea is to implement some fundamental identities in symmetric function
Symmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
theory, at the level of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s or other representing object in more abstract theories. Here k ≥ 0 is a given integer.
The fundamental idea is that for a vector bundle V on a topological space X, we should have
- ψk(V) is to Λk(V)
as
- the power sumPower sumThe topic of power sums is treated at:* Power sum symmetric polynomial * Newton's identities * Symmetric function * Faulhaber's formula...
Σ αk is to the k-th elementary symmetric function σk
of the roots α of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
P(t). (Cf. Newton's identities
Newton's identities
In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials...
.) Here Λk denotes the k-th exterior power. From classical algebra it is known that the power sums are certain integral polynomials Qk in the σk. The idea is to apply the same polynomials to the Λk(V), taking the place of σk. This calculation can be defined in a K-group, in which vector bundles may be formally combined by addition, subtraction and multiplication (tensor product
Tensor 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 polynomials here are called Newton polynomials (not, however, the Newton polynomial
Newton polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...
s of interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
theory).
Justification of the expected properties comes from the line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
case, where V is a Whitney sum of line bundles. For that case treating the line bundle direct factors formally as roots is something rather standard in algebraic topology (cf. the Leray–Hirsch theorem). In general a mechanism for reducing to that case comes from the splitting principle
Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct...
for vector bundles.
Adams operations can be defined more generally in any λ-ring over the rational numbers.