In stable homotopy theory, a **ring spectrum** is a spectrum *E* together with a multiplication map

*μ*:*E*∧*E*→*E*

and a unit map

*η*:*S*→*E*,

where *S* is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is,

*μ*(id ∧*μ*) ∼*μ*(*μ*∧ id)

and

*μ*(id ∧*η*) ∼ id ∼*μ*(*η*∧ id).

Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory.

In mathematics, an **associative algebra** is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give *A* the structure of a ring; the addition and scalar multiplication operations together give *A* the structure of a vector space over *K*. In this article we will also use the term *K*-algebra to mean an associative algebra over the field *K*. A standard first example of a *K*-algebra is a ring of square matrices over a field *K*, with the usual matrix multiplication.

In mathematics, an **algebra over a field** is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, a **Hopf algebra**, named after Heinz Hopf, is a structure that is simultaneously a algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

**Algebraic K-theory** is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called

In mathematics, an **H-space**, or a **topological unital magma**, is a topological space *X* together with a continuous map μ : *X* × *X* → *X* with an identity element *e* such that μ(*e*, *x*) = μ(*x*, *e*) = *x* for all *x* in *X*. Alternatively, the maps μ(*e*, *x*) and μ(*x*, *e*) are sometimes only required to be homotopic to the identity, sometimes through basepoint preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes. Every topological group is an H-space; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.

In mathematics, **complex cobordism** is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.

In algebraic topology, a branch of mathematics, a **spectrum** is an object representing a generalized cohomology theory. This means given a cohomology theory

In the mathematical field of algebraic topology, the **homotopy groups of spheres** describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

In mathematics, the **classifying space for the unitary group** U(*n*) is a space BU(*n*) together with a universal bundle EU(*n*) such that any hermitian bundle on a paracompact space *X* is the pull-back of EU(*n*) by a map *X* → BU(*n*) unique up to homotopy.

In mathematics, **stable homotopy theory** is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example,

In mathematics, **Brown–Peterson cohomology** is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime *p*. It is described in detail by Douglas Ravenel . Its representing spectrum is denoted by BP.

In algebraic geometry and algebraic topology, branches of mathematics, **A**^{1}**homotopy theory** is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line **A**^{1}, which is. The theory requires a substantial amount of technique to set up, but has spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.

**Michael Jerome Hopkins** is an American mathematician known for work in algebraic topology.

In stable homotopy theory, a branch of mathematics, the **sphere spectrum***S* is the monoidal unit in the category of spectra. It is the suspension spectrum of *S*^{0}, i.e., a set of two points. Explicitly, the *n*th space in the sphere spectrum is the *n*-dimensional sphere *S*^{n}, and the structure maps from the suspension of *S*^{n} to *S*^{n+1} are the canonical homeomorphisms. The *k*-th homotopy group of a sphere spectrum is the *k*-th stable homotopy group of spheres.

In mathematics, **assembly maps** are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

In mathematics, a **highly structured ring spectrum** or -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an -ring is called an -ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.

In algebra, a **simplicial commutative ring** is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If *A* is a simplicial commutative ring, then it can be shown that is a commutative ring and are modules over that ring

This is a glossary of properties and concepts in algebraic topology in mathematics.

- Adams, J. Frank (1974),
*Stable homotopy and generalised homology*, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-00523-2, MR 0402720

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