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43 relations: Abelian group, Algebraic topology, Bott periodicity theorem, Category (mathematics), Classifying space, Cohomology, Connective spectrum, CW complex, Edwin Spanier, Eilenberg–MacLane space, Elon Lages Lima, Frank Adams, G-spectrum, George W. Whitehead, Grothendieck group, Highly structured ring spectrum, Homotopy category, K-theory spectrum, List of cohomology theories, London Mathematical Society, Mapping cone (topology), Mapping spectrum, Mathematical Proceedings of the Cambridge Philosophical Society, Mathematics, Michael Atiyah, Michael Boardman, Module spectrum, Monoid, Monoidal category, N-sphere, Ring (mathematics), Ring spectrum, Simplicial set, Smash product, Sphere spectrum, Stable homotopy theory, Suspension (topology), Symmetric spectrum, Topological K-theory, Transactions of the American Mathematical Society, Triangulated category, Unitary group, Vector bundle.
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 the order in which they are written.
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Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
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Bott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
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Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.
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Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG.
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Cohomology
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.
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Connective spectrum
In algebraic topology, a branch of mathematics, a connective spectrum is a spectrum whose homotopy sets \pi_k of negative degrees are zero.
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CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.
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Edwin Spanier
Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology.
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Eilenberg–MacLane space
In mathematics, and algebraic topology in particular, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name.
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Elon Lages Lima
Elon Lages Lima (July 9, 1929 – May 7, 2017) was a Brazilian mathematician whose research concerned differential topology, algebraic topology, and differential geometry.
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Frank Adams
John Frank Adams FRS (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory.
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G-spectrum
In algebraic topology, a G-spectrum is a spectrum with an action of a (finite) group.
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George W. Whitehead
George William Whitehead, Jr. (August 2, 1918 – April 12, 2004) was an American professor of mathematics at the Massachusetts Institute of Technology, a member of the United States National Academy of Sciences, and a Fellow of the American Academy of Arts and Sciences.
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Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid M in the most universal way in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck–Riemann–Roch theorem.
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Highly structured ring spectrum
In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory.
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Homotopy category
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape.
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K-theory spectrum
In mathematics, given a ring R, the K-theory spectrum of R is an Ω-spectrum K_R whose n-th term is given by, writing \Sigma R for the suspension of R, where "+" means the Quillen's + construction.
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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.
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London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS) and the Institute of Mathematics and its Applications (IMA)).
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Mapping cone (topology)
In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology, analogous to a quotient space.
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Mapping spectrum
In algebraic topology, the mapping spectrum F(X, Y) of spectra X, Y is characterized by.
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Mathematical Proceedings of the Cambridge Philosophical Society
Mathematical Proceedings of the Cambridge Philosophical Society is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Michael Atiyah
Sir Michael Francis Atiyah (born 22 April 1929) is an English mathematician specialising in geometry.
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Michael Boardman
John Michael Boardman is a mathematician whose speciality is algebraic and differential topology.
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Module spectrum
In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra.
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Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
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Monoidal category
In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.
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N-sphere
In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring spectrum
In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map and a unit map where S is the sphere spectrum.
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Simplicial set
In mathematics, a simplicial set is an object made up of "simplices" in a specific way.
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Smash product
In mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y is the quotient of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x ∈ X and y ∈ Y.
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Sphere spectrum
In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the initial object in the category of spectra.
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Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.
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Suspension (topology)
In topology, the suspension SX of a topological space X is the quotient space: of the product of X with the unit interval I.
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Symmetric spectrum
In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group \Sigma_n on X_n such that the composition of structure maps is equivariant with respect to \Sigma_p \times \Sigma_n.
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Topological K-theory
In mathematics, topological -theory is a branch of algebraic topology.
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Transactions of the American Mathematical Society
The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.
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Triangulated category
In mathematics, a triangulated category is a category together with the additional structure of a "translation functor" and a class of "distinguished triangles".
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Unitary group
In mathematics, the unitary group of degree n, denoted U(n), is the group of unitary matrices, with the group operation of matrix multiplication.
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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 (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Eilenberg-MacLane spectrum, Eilenberg-Maclane spectrum, Eilenberg–MacLane spectrum, Homotopy spectrum, Spectrum (homotopy theory), Spectrum of spaces, Stable homotopy category, Ω-spectrum.
References
[1] https://en.wikipedia.org/wiki/Spectrum_(topology)
