View source for K-theory

A part of [[Algebraic topology|algebraic topology]] that studies properties of vector bundles by algebraic and topological methods. As opposed to [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550302.png" />-theory]], it is sometimes called topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550304.png" />-theory. In a wide sense, the term  "K-theory"  is used to denote the branch of mathematics that includes algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550305.png" />-theory and topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550306.png" />-theory, and it is characterized by specific algebraic and topological methods of investigation, which are called methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550307.png" />-theory. In the narrow sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550308.png" />-theory is the generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]) generated by the category of vector bundles (cf. [[Vector bundle|Vector bundle]]).

The source of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k0550309.png" />-theory is to be found in the skew-vector products (bundles) studied in algebraic topology, as well as in their numerous homotopic and algebraic properties. The most important properties of bundles and of concepts used in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503010.png" />-theory are the properties of characteristic classes (cf. [[Characteristic class|Characteristic class]]) of bundles, of classifying spaces (cf. [[Classifying space|Classifying space]]), of algebraic operations with bundles (direct sums, tensor products, exterior degrees), and of the inverse image of a bundle. A second source of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503011.png" />-theory is the connection with algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503012.png" />-theory, which consists of the fact that the space of continuous sections of a vector bundle can be regarded as a module over the algebra of continuous functions, which turns out to be a [[Projective module|projective module]].

In analogy with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503013.png" />-functors in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503014.png" />-theory, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503015.png" /> were defined as the Grothendieck groups (cf. [[Grothendieck group|Grothendieck group]]) of the category of vector bundles with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503016.png" /> as base. Using the concept of an [[Induced fibre bundle|induced fibre bundle]], the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503017.png" /> are completed to define a functor from the category of topological spaces into the category of Abelian groups. Usually a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503018.png" />-functor is studied not on the whole category of topological spaces, but on smaller subcategories. The category most often used is that of cellular spaces (complexes, cf. [[CW-complex|CW-complex]]). The definition of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503019.png" />-functor is extended to the category of pointed topological spaces and pairs of topological spaces and yields the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503021.png" />, by putting

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503023.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503024.png" />-dimensional disc and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503025.png" /> is its boundary. The set of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503027.png" />, satisfies the axioms of generalized cohomology theory, which should be altered by taking into account the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503028.png" />.

One distinguishes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503029.png" />-theory built on the category of real vector bundles (real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503031.png" />-theory) from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503032.png" />-theory built on complex vector bundles (complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503034.png" />-theory). Other variants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503035.png" />-theory are also studied, which take into account additional structures in certain bundles, for instance equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503037.png" />-theory.

A property of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503038.png" /> which is important in the construction of generalized cohomology theory is Bott periodicity in complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503039.png" />-theory (cf. [[Bott periodicity theorem|Bott periodicity theorem]]). It allows one, in particular, to get rid of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503040.png" /> and to transform the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503041.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503042.png" />-graded generalized cohomology theory. The fundamental importance of Bott periodicity is due to the vast computational possibilities of the constructed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503043.png" />-theory.

The computational methods of generalized cohomology theory are applicable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503044.png" />-theory, among them, in particular, the method of spectral sequences (cf. [[Spectral sequence|Spectral sequence]]), allowing one to compute the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503045.png" /> for many classical finite-dimensional and infinite-dimensional spaces. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503046.png" /> is a complex projective space, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503047.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503049.png" /> is the one-dimensional canonical bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503050.png" />.

For real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503051.png" />-theory, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503052.png" />-graded generalized cohomology theory is constructed using the [[Bott periodicity theorem|Bott periodicity theorem]]. Applying supplementary algebraic structures in vector bundles, more general cohomology theories were constructed, organically containing complex, real, and symplectic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503053.png" />-theories.

In the 1960s, with the help of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503054.png" />-theory many problems both in topology and in other branches of mathematics were reconsidered. The most important results in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503055.png" />-theory were connected with the systematic study of characteristic classes and of cohomology operations (cf. [[Cohomology operation|Cohomology operation]]) in terms of which theorems about the integral nature for multiplicative genera (analogues of the [[Riemann–Roch theorem|Riemann–Roch theorem]]) were proved, and simple and lucid solutions were given for classical problems connected with division algebras and with vector fields on spheres.

The methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503056.png" />-theory were the origin for the development of many branches of topology, e.g. the theory of bordism. The proof of the Adams' conjecture on the description of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503057.png" />-functor in terms of cohomology operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503058.png" />-theory was of fundamental importance for the development of algebraic topology (see [[#References|[2]]]).

Most impressive is the application of methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503059.png" />-theory to the problem of computing the indices of elliptic operators. With the help of geometric constructions of vector bundles, far-reaching generalizations were obtained of the concepts of a differential and a [[Pseudo-differential operator|pseudo-differential operator]], of their symbols, of a [[Sobolev space|Sobolev space]], and of elliptic operators and their indices. Also, the Atiyah–Singer formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503060.png" /></td> </tr></table>

was obtained [[#References|[3]]]; it describes the index of an elliptic operator with symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503061.png" /> on a compact closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503062.png" /> in terms of the [[Todd class|Todd class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503063.png" /> and the [[Chern character|Chern character]] of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503064.png" />. As corollaries to the Atiyah–Singer formula many special formulas have been obtained for various classes of operators which are most important in geometry, topology and other branches of mathematics. For instance, Hirzebruch's formula expresses the [[Signature|signature]] of an oriented compact, closed manifold in terms of the characteristic [[Pontryagin number|Pontryagin number]] of this manifold. The Hirzebruch formula itself and its generalizations to non-simply-connected manifolds is applied in differential topology to problems on the classification of smooth structures of manifolds.

In the 1970s generalizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503065.png" />-theory connected with the applications in it of functional-analytic methods and with the adaptation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503066.png" />-theory to many problems of topology, geometry and of the theory of differential equations appeared. One of the generalizations consists of the replacement of the category of vector bundles by that of locally trivial fibre bundles (cf. [[Locally trivial fibre bundle|Locally trivial fibre bundle]]) in which the fibres are finitely-generated modules which are projective over some [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503067.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503068.png" /> and the structure groups of which are groups of automorphisms of these modules. Using this class of fibre bundles, non-trivial cohomology invariants were constructed for infinite-dimensional representations of infinite discrete groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503069.png" />. If a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503070.png" /> is the fundamental group of a compact manifold admitting a Riemannian metric with non-positive two-dimensional curvature, then the characteristic numbers of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503071.png" /></td> </tr></table>

that is, the higher signatures, are homology invariants. (Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503072.png" /> is an arbitrary rational Pontryagin–Hirzebruch class of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503073.png" /> with fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503074.png" />.)

In a wider sense, the methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503075.png" />-theory greatly influenced the development of ideas in differential topology. With the help of a combination of algebraic and topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503076.png" />-theories, problems in differential topology were solved about the classification of smooth and of piecewise-linear structures on manifolds, about the homological and topological invariance of the characteristic Pontryagin classes, etc. Methods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503077.png" />-theory have extensive applications in functional analysis, and in particular in the theory of Banach algebras (cf. [[Banach algebra|Banach algebra]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Atiyah,   F. Hirzebruch,   "Vector bundles and homogeneous spaces"  C.B. Allendoerfer (ed.) , ''Differential geometry'' , ''Proc. Symp. Pure Math.'' , '''3''' , Amer. Math. Soc.  (1961)  pp. 7–38</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Sullivan,   "Genetics of homotopy theory"  ''Ann. of Math.'' , '''100'''  (1974)  pp. 1–79</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators on compact manifolds"  ''Bull. Amer. Math. Soc.'' , '''69'''  (1963)  pp. 422–433</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.F. Atiyah,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503078.png" />-theory: lectures" , Benjamin  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M. Karoubi,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503079.png" />-theory: an introduction" , Springer  (1978)  (Translated from French)</TD></TR></table>



====Comments====
A most useful formula for calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503080.png" /> groups is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503081.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503082.png" />-dimensional vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503083.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503084.png" /> the associated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503085.png" /> whose fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503086.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503087.png" />-dimensional projective space of lines in the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503089.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503090.png" />. Over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503091.png" /> there is the canonical line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503092.png" />, whose fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503093.png" />  "is"  the one-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503094.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503096.png" />, a one-point space, this is the canonical line bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503098.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k05503099.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030100.png" /> be the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030101.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030102.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030103.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030104.png" />-module with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030105.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030106.png" /> is given by the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030107.png" /></td> </tr></table>

In equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030108.png" />-theory one considers spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030109.png" /> with an action of a (Lie) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030110.png" /> on it. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030112.png" />-vector bundle is a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030113.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030114.png" /> acts on both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030117.png" /> is a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030118.png" />-spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030119.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030120.png" /> by vector bundle automorphisms (i.e. the restrictions to the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030121.png" /> are linear). One has direct sums and tensor products of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030122.png" />-vector bundles, yielding a semi-ring of isomorphism classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030123.png" />. The equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030124.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030125.png" /> for compact spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030126.png" /> is now the [[Grothendieck group|Grothendieck group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030127.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030129.png" />, the representation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030130.png" />. For a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030131.png" /> with base point, the reduced groups are defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030132.png" />. For locally compact non-compact spaces (with proper <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030133.png" />-action), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030134.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030135.png" /> is the one-point compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030136.png" />. The higher groups are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030138.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030139.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030140.png" /> is compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030141.png" /> is the one-point compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030142.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030143.png" /> is locally compact but not compact, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030144.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030145.png" />-fold suspension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030147.png" /> is the cone over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030148.png" />.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030149.png" /> thus defined form again a generalized cohomology theory. Cf. [[#References|[a1]]], [[#References|[a6]]]–[[#References|[a9]]] for more details and applications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030150.png" />.

For a survey of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030151.png" />-algebra generalizations and ramifications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030152.png" />-theory (in particular, for an account of Kasparov <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030153.png" />-theory) see [[#References|[a2]]]. In addition, an extremely important connection between the topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030154.png" />-theory and the algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030155.png" />-theory of smooth complex varieties has been found by R.W. Thomason (see [[#References|[a3]]], [[#References|[a4]]]). Essentially, Thomason showed that the algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030156.png" />-theory of a smooth variety, when taken with finite coefficients, becomes topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030157.png" />-theory if one localizes in such a way as to impose Bott periodicity. More recently, interrelations have been discovered between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030158.png" />-theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030159.png" />-algebras, Connes' cyclic homology and Waldhausen's algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030160.png" />-theory of spaces. For details see [[#References|[a5]]], for example.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030161.png" />-theory and representation theory are directly related by the fact that for a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030162.png" /> the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030163.png" />-theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030164.png" /> of the classifying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030165.png" /> is the completion of the complex representation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030166.png" /> for the topology defined by the augmentation ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030167.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030168.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030169.png" />, [[#References|[a10]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.B. Segal,   "Equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030170.png" />-theory"  ''Publ. Math. IHES'' , '''34'''  (1968)  pp. 129–151</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Blackadar,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030171.png" />-theory for operator algebras" , Springer  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.C. Dwyer,   E.M. Friedlander,   V.P. Snaith,   R.W. Thomason,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030172.png" />-theory eventually surjects on topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030173.png" />-theory"  ''Invent. Math.'' , '''66'''  (1982)  pp. 481–491</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.W. Thomason,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030174.png" />-theory and étale cohomology"  ''Ann. Sci. Ec. Norm. Sup.'' , '''18'''  (1985)  pp. 437–552</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T.G. Goodwillie,   "On the general linear group and Hochshild homology"  ''Ann. of Math. (2)'' , '''121'''  (1985)  pp. 383–407</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T. Petrie,   J.D. Randall,   "Transformation groups on manifolds" , M. Dekker  (1984)  pp. 8, 9</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M.F. Atiyah,   G. Segal,   "The index of elliptic operators II"  ''Ann. of Math.'' , '''87'''  (1968)  pp. 531–545</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Illusie,   "Nombres de Chern et groupes finis"  ''Topology'' , '''7'''  (1968)  pp. 255–270</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.F. Atiyah,   "Lectures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030175.png" />-theory" , Math. Inst. Harvard Univ.  (1964)  (Mimeographed notes)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M.F. Atiyah,   G. Segal,   "Equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055030/k055030176.png" />-theory and completion"  ''J. Diff. Geom.'' , '''3'''  (1969)  pp. 1–18</TD></TR></table>