K-theory

A part of algebraic topology that studies properties of vector bundles by algebraic and topological methods. As opposed to algebraic $ K $-theory, it is sometimes called topological $ K $-theory. In a wide sense, the term "K-theory" is used to denote the branch of mathematics that includes algebraic $ K $-theory and topological $ K $-theory, and it is characterized by specific algebraic and topological methods of investigation, which are called methods of $ K $-theory. In the narrow sense, $ K $-theory is the generalized cohomology theory (cf. Generalized cohomology theories) generated by the category of vector bundles (cf. Vector bundle).

The source of $ K $-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 $ K $-theory are the properties of characteristic classes (cf. Characteristic class) of bundles, of classifying spaces (cf. 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 $ K $-theory is the connection with algebraic $ K $-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.

In analogy with $ K $-functors in algebraic $ K $-theory, the groups $ K ( X) $ were defined as the Grothendieck groups (cf. Grothendieck group) of the category of vector bundles with $ X $ as base. Using the concept of an induced fibre bundle, the groups $ K ( X) $ are completed to define a functor from the category of topological spaces into the category of Abelian groups. Usually a $ K $-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). The definition of a $ K $-functor is extended to the category of pointed topological spaces and pairs of topological spaces and yields the groups $ K ^ {i} ( X, A) $, $ i \leq 0 $, by putting

$$ K ^ {i} ( X, A) = \ K ( X \times D ^ {-1} ,\ X \times S ^ {- i - 1 } \cup A \times D ^ {-i} ), $$

where $ D ^ {-i} $ is an $ i $-dimensional disc and $ S ^ {- i - 1 } $ is its boundary. The set of functors $ K ^ {i} $, $ i \leq 0 $, satisfies the axioms of generalized cohomology theory, which should be altered by taking into account the inequality $ i \leq 0 $.

One distinguishes the $ K $-theory built on the category of real vector bundles (real $ K $-theory) from the $ K $-theory built on complex vector bundles (complex $ K $-theory). Other variants of $ K $-theory are also studied, which take into account additional structures in certain bundles, for instance equivariant $ K $-theory.

A property of the group $ K ( X) $ which is important in the construction of generalized cohomology theory is Bott periodicity in complex $ K $-theory (cf. Bott periodicity theorem). It allows one, in particular, to get rid of the restriction $ i \leq 0 $ and to transform the functors $ K ^ {i} $ into a $ \mathbf Z _ {2} $-graded generalized cohomology theory. The fundamental importance of Bott periodicity is due to the vast computational possibilities of the constructed $ K $-theory.

The computational methods of generalized cohomology theory are applicable to $ K $-theory, among them, in particular, the method of spectral sequences (cf. Spectral sequence), allowing one to compute the groups $ K ( X) $ for many classical finite-dimensional and infinite-dimensional spaces. For instance, if $ X = \mathbf C P ^ {n} $ is a complex projective space, then

$$ K ( \mathbf C P ^ {n} ) = \ \mathbf Z [ u] / ( u ^ {n + 1 } ) , $$

where $ u = [ \eta ] - 1 $, and $ \eta $ is the one-dimensional canonical bundle over $ \mathbf C P ^ {n} $.

For real $ K $-theory, a $ \mathbf Z _ {8} $-graded generalized cohomology theory is constructed using the Bott periodicity theorem. Applying supplementary algebraic structures in vector bundles, more general cohomology theories were constructed, organically containing complex, real, and symplectic $ K $-theories.

In the 1960s, with the help of $ K $-theory many problems both in topology and in other branches of mathematics were reconsidered. The most important results in $ K $-theory were connected with the systematic study of characteristic classes and of cohomology operations (cf. Cohomology operation) in terms of which theorems about the integral nature for multiplicative genera (analogues of the 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 $ K $-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 $ J $-functor in terms of cohomology operations in $ K $-theory was of fundamental importance for the development of algebraic topology (see [2]).

Most impressive is the application of methods of $ K $-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, of their symbols, of a Sobolev space, and of elliptic operators and their indices. Also, the Atiyah–Singer formula

$$ \mathop{\rm ind} \sigma ( D) = \ \langle \mathop{\rm ch} \sigma \cdot T ( X), [ X] \rangle $$

was obtained [3]; it describes the index of an elliptic operator with symbol $ \sigma $ on a compact closed manifold $ X $ in terms of the Todd class $ T ( X) $ and the Chern character of the operator $ \sigma ( D) $. 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 of an oriented compact, closed manifold in terms of the characteristic 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 $ K $-theory connected with the applications in it of functional-analytic methods and with the adaptation of $ K $-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) in which the fibres are finitely-generated modules which are projective over some $ C ^ {*} $-algebra $ A $ 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 $ \pi $. If a discrete group $ \pi $ 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

$$ \tau _ {x} ( M) = \langle L ( M) x, [ M] \rangle, $$

that is, the higher signatures, are homology invariants. (Here, $ x $ is an arbitrary rational Pontryagin–Hirzebruch class of the manifold $ M $ with fundamental group $ \pi $.)

In a wider sense, the methods of $ K $-theory greatly influenced the development of ideas in differential topology. With the help of a combination of algebraic and topological $ K $-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 $ K $-theory have extensive applications in functional analysis, and in particular in the theory of Banach algebras (cf. Banach algebra).

References

[1]	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 MR0139181 Zbl 0108.17705
[2]	D. Sullivan, "Genetics of homotopy theory" Ann. of Math. , 100 (1974) pp. 1–79 MR0442930 Zbl 0355.57007
[3]	M.F. Atiyah, I.M. Singer, "The index of elliptic operators on compact manifolds" Bull. Amer. Math. Soc. , 69 (1963) pp. 422–433 MR0157392 Zbl 0118.31203
[4]	N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) MR0039258 Zbl 0054.07103
[5]	M.F. Atiyah, "-theory: lectures" , Benjamin (1967) MR224083
[6]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[7]	M. Karoubi, "-theory: an introduction" , Springer (1978) (Translated from French) MR488029 Zbl 0382.55002

Comments

A most useful formula for calculating $ K $ groups is the following. Let $ E $ be an $ n $-dimensional vector bundle over $ X $. Denote by $ \mathbf P ( E) $ the associated $ \mathbf C P _ {n-} 1 ( \mathbf C ) $ whose fibre over $ x \in X $ is the $ ( n - 1 ) $-dimensional projective space of lines in the fibre $ E _ {x} $ of $ E $ over $ x \in X $. Over $ \mathbf P ( E) $ there is the canonical line bundle $ \xi _ {E} $, whose fibre over $ y \in \mathbf P ( E ) $" is" the one-dimensional vector space $ y $. For $ E = \mathbf C ^ {n+1} $, $ X = \{ \mathop{\rm pt} \} $, a one-point space, this is the canonical line bundle over $ P _ {n} ( \mathbf C ) = \mathbf C P ^ {n} $ given by $ \xi _ {n} = \{ {( x , v) \in P _ {n} ( \mathbf C ) \times \mathbf C ^ {n+1} } : {v \in x } \} $. Let $ t = [ \xi _ {E} ] $ be the class of $ \xi _ {E} $ in $ K ( \mathbf P ( E) ) $. Then $ K ^ {*} ( \mathbf P ( E) ) $ is a free $ K ^ {*} ( X) $-module with basis $ 1 , t \dots t ^ {n-1} $, and $ t ^ {n} $ is given by the relation

$$ t ^ {n} - [ \lambda ^ {1} ( E) ] t ^ {n-1} + \dots + ( - 1 ) ^ {n} [ \lambda ^ {n} ( E) ] = 0 . $$

In equivariant $ K $-theory one considers spaces $ X $ with an action of a (Lie) group $ G $ on it. A $ G $-vector bundle is a vector bundle $ \pi : E \rightarrow X $ where $ G $ acts on both $ E $ and $ X $ and $ \pi $ is a mapping of $ G $-spaces and $ G $ acts on $ E $ by vector bundle automorphisms (i.e. the restrictions to the fibres $ g : E _ {x} \rightarrow E _ {gx} $ are linear). One has direct sums and tensor products of $ G $-vector bundles, yielding a semi-ring of isomorphism classes $ \mathop{\rm Vect} _ {G} ( X) $. The equivariant $ K $-group $ K _ {G} ( X) $ for compact spaces $ X $ is now the Grothendieck group of $ \mathop{\rm Vect} _ {G} ( X) $.

For $ X = \mathop{\rm pt} $, $ K _ {G} ( \mathop{\rm pt} ) = R( G) $, the representation ring of $ G $. For a space $ ( X , x _ {0} ) $ with base point, the reduced groups are defined as $ \widetilde{K} _ {G} ( X) = \mathop{\rm Ker} ( K _ {G} ( X) \rightarrow K _ {G} ( x _ {0} ) ) $. For locally compact non-compact spaces (with proper $ G $-action), $ K _ {G} ( X) = \widetilde{K} _ {G} ( X ^ {+} ) $, where $ X ^ {+} $ is the one-point compactification of $ X $. The higher groups are defined by $ K _ {G} ^ {-n} ( X) = \widetilde{K} {} _ {G} ^ {-n} ( X ^ {+} ) = \widetilde{K} _ {G} ( S ^ {n} ( X ^ {+} ) ) $, $ K _ {G} ^ {-n} ( X , A ) = \widetilde{K} _ {G} ( S ^ {n} ( X ^ {+} \cup _ {A ^ {+} } C A ^ {+} ) ) $, where $ X ^ {+} = X \cup \{ \mathop{\rm pt} \} $ if $ X $ is compact and $ X ^ {+} $ is the one-point compactification of $ X $ if $ X $ is locally compact but not compact, and where $ S ^ {n} Y $ is the $ n $-fold suspension of $ Y $ and $ C Z $ is the cone over $ Z $.

The $ K _ {G} ^ {*} ( X , A ) $ thus defined form again a generalized cohomology theory. Cf. [a1], [a6]–[a9] for more details and applications of $ K _ {G} ^ {*} $.

For a survey of the $ C ^ {*} $-algebra generalizations and ramifications of $ K $-theory (in particular, for an account of Kasparov $ KK $-theory) see [a2]. In addition, an extremely important connection between the topological $ K $-theory and the algebraic $ K $-theory of smooth complex varieties has been found by R.W. Thomason (see [a3], [a4]). Essentially, Thomason showed that the algebraic $ K $-theory of a smooth variety, when taken with finite coefficients, becomes topological $ K $-theory if one localizes in such a way as to impose Bott periodicity. More recently, interrelations have been discovered between the $ K $-theory of $ C ^ {*} $-algebras, Connes' cyclic homology and Waldhausen's algebraic $ K $-theory of spaces. For details see [a5], for example.

$ K $-theory and representation theory are directly related by the fact that for a compact group $ G $ the complex $ K $-theory $ K ^ {*} ( B G ) $ of the classifying space $ B G $ is the completion of the complex representation ring $ R ( G) $ for the topology defined by the augmentation ideal $ I _ {G} $ of $ R ( G) $: $ K ^ {*} ( B G ) = \widehat{ {R ( G) }} $, [a10].

References

[a1]	G.B. Segal, "Equivariant -theory" Publ. Math. IHES , 34 (1968) pp. 129–151
[a2]	B. Blackadar, "-theory for operator algebras" , Springer (1986) MR0859867 Zbl 0597.46072
[a3]	W.C. Dwyer, E.M. Friedlander, V.P. Snaith, R.W. Thomason, "Algebraic -theory eventually surjects on topological -theory" Invent. Math. , 66 (1982) pp. 481–491 MR662604
[a4]	R.W. Thomason, "Algebraic -theory and étale cohomology" Ann. Sci. Ec. Norm. Sup. , 18 (1985) pp. 437–552
[a5]	T.G. Goodwillie, "On the general linear group and Hochshild homology" Ann. of Math. (2) , 121 (1985) pp. 383–407
[a6]	T. Petrie, J.D. Randall, "Transformation groups on manifolds" , M. Dekker (1984) pp. 8, 9 MR0748850 Zbl 0574.57001
[a7]	M.F. Atiyah, G. Segal, "The index of elliptic operators II" Ann. of Math. , 87 (1968) pp. 531–545 MR0236953 MR0236951 Zbl 0164.24201
[a8]	L. Illusie, "Nombres de Chern et groupes finis" Topology , 7 (1968) pp. 255–270 MR0229248
[a9]	M.F. Atiyah, "Lectures on -theory" , Math. Inst. Harvard Univ. (1964) (Mimeographed notes)
[a10]	M.F. Atiyah, G. Segal, "Equivariant -theory and completion" J. Diff. Geom. , 3 (1969) pp. 1–18 MR259946