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Equivariant K-theory of real vector spaces and real projective spaces
... For this purpose, we use the "Chern character" for finite group actions, as defined by Baum, Connes, Kuhn and Slominska [11][32] [40], together with our generalized Thom isomorphism. These last computations are related to some previous ones [30] and to the work of many authors. Finally, we introduce new cohomology operations which are complementary to those defined in [19] and [9]. ...
... Therefore, G is a central covering of G with fiber µ n (whose elements are denoted by Greek letters such as λ). The following definition is already present in [30] §2.5 (for n = 2): ...
... G (X) is canonically isomorphic to the Grothendieck group of the category E G (X) l Proof. One just repeats the argument in the proof of Theorem 2.6 in [30], where A is a Clifford algebra C(V ) and Z/2 plays the role of µ n . We simply "untwist" the action of G thanks to the formula (F) written explicitly in the proof of 2.6 (loc. ...
Twisted K-theory has its origins in the author's PhD thesis [27] : http://www.numdam.org/item?id=ASENS_1968_4_1_2_161_0 and in the paper with P. Donovan http://www.numdam.org/item?id=PMIHES_1970__38__5_0 The objective of this paper is to revisit the subject in the light of generalizations and new developments inspired by Mathematical Physics. See for instance E. Witten (hep-th/9810188), J. Rosenberg http://anziamj.austms.org.au/JAMSA/V47/Part3/Rosenberg.html, C. Laurent-Gentoux, J.-L. Tu, P. Xu (math/0306138) and M.F. Atiyah, G. Segal (math/0407054), among many authors. The unifiyng theme in our presentation is the notion of K-theory of graded Banach algebras,implicit in [27], from which most of the classical theorems in twisted K-theory are derived. We also prove some new results in the subject : a Thom isomorphism in this setting, explicit computations in the equivariant case and new cohomology operations (in the graded and ungraded cases).
... In general however, we do not have a complete recipe for π * λ KU G . As a group, π λ KU G is free abelian of known finite rank by work of Karoubi [Kar02], and one always has Bott periodicity in the form π ( * +2)λ KU G ∼ = π * λ KU G {β C⊗λ }. Thus we are left with the following problem. ...
... Here, the product is over the conjugacy classes of elements g ∈ G, and C g is the centralizer of g acting on the fixed points S α g . This is used to describe the rank of π α KU G in [Kar02]. In particular, the sequence of Problem 4.2.1 is easily understood after complexification. ...
... The composite is determined by a 2 λ β λ C = a λ C β λ C = e λ C = 1 − λ C + σ C , and has image the rank 1 subspace Z{1 − λ C + σ C } ⊂ RU (Σ 3 ). As π λ KU Σ3 is a free abelian group of rank 1 [Kar02], the only possibility is that π λ KU Σ3 ∼ = Z{a λ β λ C } with σ C a λ = a λ and λ C a λ = −a λ . Thus π * λ KU Σ3 has the 2-periodic pattern ...
For a finite group G and virtual G-representation of virtual dimension 0, there is an invertible Thom class in the RO(G)-graded coefficients of G-equivariant cobordism. We introduce and study -self maps: equivalences inducing multiplication by in -theory. We also treat the variants based on and , as well as equivalences not necessarily compatible with cobordism. When arises as the cofiber of an Euler class, these periodicities may be produced by an RO(G)-graded J-homomorphism , and we use this to give several examples.
... For B = C 0 (R) however, they are nothing but the equivariant topological Ktheory K * H R H for the H -Euclidean space R H , i.e., R |H | with H -action induced by translation of coordinates. The groups K * H R H are quite well-studied [10,15]. Using these results, we give a more explicit formula for A = C(S 1 ) (Example 4.2) and for the rotation algebras (or noncommutative tori) A = A θ (Example 4.4). ...
... In general, for any finite group H and for any orthogonal representation H → O(V ) on a finite-dimensional Euclidean space V , the K -theory of C 0 (V ) r H is the wellstudied equivariant topological K -theory K * H (V ) of V (see [10,15]). It is known to be a finitely generated free abelian group with rank Z K * H (V ) equal to the number of conjugacy classes g of H which are oriented and even/odd respectively (see [15,Theorem 1.8]). ...
... In general, for any finite group H and for any orthogonal representation H → O(V ) on a finite-dimensional Euclidean space V , the K -theory of C 0 (V ) r H is the wellstudied equivariant topological K -theory K * H (V ) of V (see [10,15]). It is known to be a finitely generated free abelian group with rank Z K * H (V ) equal to the number of conjugacy classes g of H which are oriented and even/odd respectively (see [15,Theorem 1.8]). Here, a conjugacy class g of H is oriented if the centralizer C g of g acts on the g-fixed points V g of V by oriented automorphisms. ...
For a large class of C∗-algebras A, we calculate the K-theory of reduced crossed products A⊗G⋊rG of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional C∗-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C∗-algebras of wreath products H≀G for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.
... (20) Here Irr Z 2 (G) are all Z 2 -graded and -twisted irreducible representations for suitable grading and twisting explicitly determined by the point-group, G; the degrees, p ρ , q ρ , are also explicitly determined. A proof of this isomorphism was given by Karoubi [127]. For further details of this isomorphism see Appendix C. ...
... Hence, the study of the G-equivariant K-theory of the torus of a signed permutations representation reduces to that of representation spheres, which is covered by Refs. [62,127]. ...
... Note that, in particular, the algebra Cl Q [G] depends only on ρ . This "separation of variables" results from a change of coordinates [62,127]. Given a real representation ρ of G, one obtains a Z 2 -grading, ρ : G → Z 2 , on G, for which g ρ − → det(ρ(g)), i.e., the parity of g ∈ G is sign of the determinant. ...
The celebrated tenfold way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of nonspatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly interacting electrons' noncrystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct ground states as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold way of Altland-Zirnbauer nonspatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold way.
... As examples show, in this case non-trvial K-groups may appear in all dimensions. The situation has been studied in case of finite groups by Karoubi in [7]. In this paper we use different methods to give a general description of K * ...
... In this section we want to study the case of finite groups in more detail. This case was already considered by Karoubi in [7], but the methods used here are different from those used by Karoubi. We first notice that it follows from Theorem 2.5 that for actions of finite groups G, the K-theory groups K * G (V ) are always finitely generated free abelian groups. ...
... As in the Pin c -case, these equations have unique solutions and, using Corollary 2.10, we obtain [7] that for any linear action ρ : G → O(V ) of a finite group G the ranks of K 0 G (V ) and K 1 G (V ) can alternatively be computed as follows: For any conjugacy class C g in G let V g denote the fixed-point set of ρ(g) in V . This space is ρ(h)-invariant for any h in the centralizer C g of g, and therefore C g acts linearly on V g for all g in G. ...
We give a general formula for the equivariant complex K-theory K ∗ G (V) of a finite dimensional real linear space V equipped with a linear action of a compact group G in terms of the representation theory of a certain double cover of G. Using this general formula, we give explicit computations in various interesting special cases. In particular, as an application we obtain explicit formulas for the K-theory of C ∗ r (GL(n, R)), the reduced group C*-algebra of GL(n, R). Let G be a compact group acting linearly on the real vector space V. In this paper we want to give explicit formulas for the complex equivarant K-theory K ∗ G (V) depending on the action of the given group G on V. By use of the positive solution of the Connes-Kasparov conjecture in [5], this will also provide explicit formulas
... For this purpose, we use the "Chern character" for finite group actions, as defined by Baum, Connes, Kuhn and Slominska [11][32] [40], together with our generalized Thom isomorphism. These last computations are related to some previous ones [30] and to the work of many authors. Finally, we introduce new cohomology operations which are complementary to those defined in [19] and [9]. ...
... Therefore, G is a central covering of G with fiber µ n (whose elements are denoted by Greek letters such as λ). The following definition is already present in [30] §2.5 (for n = 2): ...
... G (X) is canonically isomorphic to the Grothendieck group of the category E G (X) l Proof. One just repeats the argument in the proof of Theorem 2.6 in [30], where A is a Clifford algebra C(V ) and Z/2 plays the role of µ n . We simply "untwist" the action of G thanks to the formula (F) written explicitly in the proof of 2.6 (loc. ...
Twisted K-theory has its origins in the author's PhD thesis [27] : http://www.numdam.org/item?id=ASENS_1968_4_1_2_161_0 and in the paper with P. Donovan http://www.numdam.org/item?id=PMIHES_1970__38__5_0 The objective of this paper is to revisit the subject in the light of generalizations and new developments inspired by Mathematical Physics. See for instance E. Witten (hep-th/9810188), J. Rosenberg http://anziamj.austms.org.au/JAMSA/V47/Part3/Rosenberg.html, C. Laurent-Gentoux, J.-L. Tu, P. Xu (math/0306138) and M.F. Atiyah, G. Segal (math/0407054), among many authors. The unifiyng theme in our presentation is the notion of K-theory of graded Banach algebras,implicit in [27], from which most of the classical theorems in twisted K-theory are derived. We also prove some new results in the subject : a Thom isomorphism in this setting, explicit computations in the equivariant case and new cohomology operations (in the graded and ungraded cases).
... In general, for any finite group H and for any orthogonal representation H → O(V ) on a finite-dimensional Euclidean space V , the K-theory of C 0 (V ) ⋊ r H is the well-studied equivariant topological K-theory K * H (V ) of V (see [Kar02], [EP09]). It is known to be a finitely generated free abelian group with rank Z K * H (V ) equal to the number of conjugacy classes g of H which are oriented and even/odd respectively (see [Kar02, Theorem 1.8]). ...
... Therefore, in Theorem 4.14, B can be taken as the direct sum of, possibly infinitely many, C and C 0 (R). The assertion follows from the fact that equivariant K-theory K * G 0 (R G 0 ) (more generally K * G 0 (V ) for any G 0 -Euclidean space V ) is a (finitely-generated) free abelian group for any finite group G 0 by [Kar02] (or [EP09]). ...
For a large class of C*-algebras A, we calculate the K-theory of reduced crossed products of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for finite-dimensional C*-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C*-algebras of wreath products for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.
... (20) Here, Irr Z2 (G) are all Z 2 -graded and -twisted irreducible representations for suitable grading and twisting explicitly determined by the point-group, G; the degrees, p ρ , q ρ , are also explicitly determined. A proof of this isomorphism was given by M. Karoubi in Ref. 123. ...
... Note that, in particular, the algebra Cl Q [G] depends only on ρ . This "separation of variables" results from a change of coordinates [62,123]. Given a real representation ρ of G, one obtains a Z 2 -grading, ρ : G → Z 2 , on G, for which g ρ − − → det(ρ(g)), i.e., the parity of g ∈ G is sign of the determinant. ...
The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly-interacting electrons' non-crystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct groundstates as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold-way of Altland-Zirnbauer non-spatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold-way.
... In the case of |G| even, we may encounter real representations. A general calculation of the groups (4.2) was given by Max Karoubi [47] (see also [48]). All the groups we will need in this paper however can be calculated from first principles by elementary means. ...
We systematically revisit the description of D-branes on orbifolds and the classification of their charges via K-theory. We include enough details to make the results accessible to both physicists and mathematicians interested in these topics. The minimally charged branes predicted by K-theory in ℤN orbifolds with N odd are only BPS. We confirm this result using the boundary state formalism for ℤ3. For ℤN × ℤN orbifolds with N odd, in the untwisted case and with discrete torsion whose class is a generator of H³(ℤ/N × ℤ/N,ℤ), we show that the K-theory classification of charges agrees with the boundary state approach, largely developed by Gaberdiel and collaborators, including the types of representation on the Chan–Paton factors.
... ). Now these groups of K-theory have no torsion, which was established by M. Karoubi in his article motivated by our works, [12]. The groups of K-theory thus can be computed thanks to the Chern isomorphism for discrete groups, by reducing to the computation of the homology groups. ...
This survey of the work of the author with several collaborators presents the way groupoids appear and can be used in index theory. We define the general tools, and apply them to the case of manifolds with corners, ending with a topological index theorem.
... Now the group K(X × RP n , X) has been described concretely in [24]p.249, Theorem 6.40 (see also, more recently, [30]). It is the middle term of an exact sequence ...
Rognes and Weibel used Voevodsky's work on the Milnor conjecture to deduce the strong Dwyer-Friedlander form of the Lichtenbaum-Quillen conjecture at the prime 2. In consequence (the 2-completion of) the classifying space for algebraic K-theory of the integers ℤ[1/2] can be expressed as a fiber product of well-understood spaces BO and BGL(double-struck F sign 3)+ over BU. Similar results are now obtained for Hermitian K-theory and the classifying spaces of the integral symplectic and orthogonal groups. For the integers ℤ[1/2], this leads to computations of the 2-primary Hermitian K-groups and affirmation of the Lichtenbaum-Quillen conjecture in the framework of Hermitian K-theory.
... where the involved rings are crossed products of G by Clifford algebras. More precise results may be found in [6]. ...
The purpose of this short paper is to make the link between the fundamental work of Atiyah, Bott and Shapiro [1] and twisted K-theory as defined by P. Donovan, J. Rosenberg and the author [2] [8] [7]. This link was implicit in the literature (for bundles over spheres as an example) but was not been explicitly defined before. The setting is the following: V is a real vector bundle on a compact space X, provided with a non degenerate quadratic form to which we associate a bundle of (real or complex) Clifford algebras denoted by C(V); the quadratic form is implicit in this notation. We denote by M(V) the Grothendieck group associated to the category of (real or complex) vector bundles provided with a structure of (twisted) Z/2-graded C(V)-module. Another way to describe M(V) is to consider the bundle V ⊕ 1, where the symbol “1 ” denotes the trivial vector bundle of rank one with a positive quadratic form. Then M(V) is just the Grothendieck group K(Λ1) of the category P(Λ1) whicho objects are finitely generated projective modules over Λ1. The notation Λn means in general Λ̂⊗C 0,n, where Λ is the ring of continuous sections of the Z/2-graded bundle C(V) and C 0,n is the Clifford algebra of R n with a positive quadratic form. Following [1], we define A(V) as the cokernel of the homomorphism induced by restriction of the scalars: A(V) = Coker[M(V ⊕ 1) → M(V)] = Coker[K(Λ2) → K(Λ1)]. Remark. Let us denote by V − the vector bundle V with the opposite quadratic
... If the transverse space V is instead a real linear G-module, then throughout one should restrict to the subring of R(G) consisting of representations associated to conjugacy classes [g] ∈ G ∨ for which the centralizer ZG(g) acts on the fixed point subspace V g by oriented automorphisms[40]. This will follow immediately from the isomorphism (4.7) below with X = V . ...
We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds. Comment: 46 pages; v2: typos corrected
We compute the RO(A)-graded coefficients of A-equivariant complex and real topological K-theory for A a finite elementary abelian 2-group, together with all products, transfers, restrictions, power operations, and Adams operations.
We present structure theorems in terms of inertial decompositions for the wreath product ring of an orbifold presented as the quotient of a smooth, closed manifold by a compact, connected Lie group acting almost freely. In particular we show that this ring admits λ-ring and Hopf algebra structures both abstractly and directly. This generalizes results known for global quotient orbifolds by finite groups.
We study the C*-algebra crossed product of a locally compact group G acting properly on a locally compact Hausdorff space X. Under some mild extra conditions, which are automatic if G is discrete or a Lie group, we describe in detail, and in terms of the action, the primitive ideal space of such crossed products as a topological space, in particular with respect to its fibring over the quotient space . We also give some results on the \K-theory of such C*-algebras. These more or less compute the \K-theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely \K-theoretic proof of a result due to Paul Baum and Alain Connes on (\K)-theory with complex coefficients of crossed products by finite groups.
We compute the K-theory, groups of the C*-algebra of the groupoid of a manifold with corners, in which the analytic index takes its values.
This chapter discusses the Chern character for discrete groups. Hj (X;Q) is the j-th Cech cohomology group of X with coefficients the rational numbers Q. The key property of this classical Chern character is that it is a rational isomorphism. Cyclic cohomology can be used to define the delocalized equivariant cohomology of X. The traditional homotopy quotient Chern character gives a map, which is always surjective, for compact X. The chapter also discusses twisted homology and K homology. K homology is the homology theory associated to the Z × BU spectrum. A concrete realization of this theory is obtained by using the K-cycle definition.
Thesis (doctoral)--L'Université de Paris, 1968. Bibliography: p. 267-268. Includes index.
- M.-F Atiyah
- R Bott
- R Shapiro
- Clifford Modules
M.-F. Atiyah, R. Bott, R. Shapiro, Clifford modules, Topology 3 (1964) 3–38.
- M Karoubi
- Algèbres De
M. Karoubi, Algèbres de Clifford et K-théorie, Ann. Sci. École Norm. Sup. (1968) 161–270.