In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, all of which give the same homotopy category. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ...

Suppose we start with a generalized cohomology theory E. This is a sequence of contravariant functors Eⁿ from topological spaces to abelian groups, one for each integer n, which satisfy all of the Eilenberg-Steenrod axioms except for the dimension axiom. By the Brown representability theorem, Eⁿ(X) is given by [X,E_n], the set of homotopy classes of maps from X to E_n, for some space E_n. The isomorphism , where ΣX is the suspension of X, gives a map . This collection of spaces E_n together with connecting maps is a spectrum. In most (but not all) constructions of spectra the adjoint maps are required to be weak equivalences or even homeomorphisms. In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... For functors in computer science, see the function object article. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, an abelian group is a commutative group, i. ... In mathematics, specifically in algebraic topology, the Eilenberg-Steenrod axioms are properties that homology theories of topological spaces have in common. ... In mathematics, Browns representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Sets, to be a representable functor. ... In topology, the suspension SX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse both ends to two points. ... In mathematics, the term adjoint applies in several situations. ... This word should not be confused with homomorphism. ...

Examples

Consider singular cohomology Hⁿ(X;A) with coefficients in an abelian group A. By Brown representability Hⁿ(X;A) is the set of homotopy classes of maps from X to K(A,n), the Eilenberg-MacLane space with homotopy concentrated in degree n. Then the corresponding spectrum HA has n'th space K(A,n); it is called the Eilenberg-MacLane spectrum. 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, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, Browns representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor. ... In mathematics, an Eilenberg-MacLane space is a special kind of topological space that is important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomological operations. ...

As a second important example, consider topological K-theory. At least for X compact, K⁰(X) is defined to be the group completion of the monoid of complex vector bundles on X. Also, K¹(X) is the group corresponding to vector bundles on the suspension of X. Topological K-theory is a generalized cohomology theory, so it gives a spectrum. The zero'th space is while the first space is U. Here U is the infinite unitary group and BU is its classifying space. By Bott periodicity we get and for all n, so all the spaces in the topological K-theory spectrum are given by either or U. There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-periodic spectrum. In mathematics, topological K-theory is a branch of algebraic topology. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In mathematics, the Bott periodicity theorem is a result from homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved to be of foundational significance for much further research, in particular in K-theory. ...

For many more examples, see the list of cohomology theories. This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. ...

History

A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. There was development of the topic by J. Michael Boardman, amongst others. The above setting came together during the mid-1960s, and is still used for many purposes: see Adams (1974) or Vogt (1970). Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses modified (and highly technical) definitions of spectrum: see Mandell et al. (2001) for a unified treatment of these new approaches. Edwin Henry Spanier (August 8, 1921, Washington, D.C.–October 11, 1996, Scottsdale, Arizona) was an American mathematician at the University of California at Berkeley, working in algebraic topology. ... John Michael Boardman is a mathematician whose speciality is algebraic and differential topology. ... Frank Adams may also refer to Frank Dawson Adams a Canadian geologist. ...

References

• J. F. Adams (1974). "Stable homotopy and generalised homology". University of Chicago Press.

• Mandell, M. A.; May,, J. P. & Schwede, S. et al. (2001), "Model categories of diagram spectra", Proc. London Math. Soc. (3) 82: 441-512, DOI 10.1112/S0024611501012692

• R. Vogt (1970). "Boardman's stable homotopy category". Lecture note series No. 21, Matematisk Institut, Aarhus University.