Complex K-theory Research Articles

K-Theory and Formality

Abstract We compare several notions of equivariant formality for complex K-theory with respect to a compact Lie group action, including surjectivity of the forgetful map and the weak equivariant formality of Harada–Landweber, and find all are equivalent under standard hypotheses. As a consequence, we present an expression for the equivariant K-theory of the isotropy action of $H$ on a homogeneous space $G/H$ in all the classical cases. The proofs involve mainly homological algebra and arguments with the Atiyah–Hirzebruch–Leray–Serre spectral sequence, but a more general result depends on a map of spectral sequences from Hodgkin’s Künneth spectral sequence in equivariant K-theory to that in Borel cohomology that seems not to have been otherwise defined. The hypotheses for the main structure result are analogous to a previously announced characterization of cohomological equivariant formality, first proved here, expanding on the results of Shiga and Takahashi.

Naive-commutative ring structure on rational equivariant K-theory for abelian groups

In this paper, we calculate the image of the connective and periodic rational equivariant complex K-theory spectrum in the algebraic model for naive-commutative ring G-spectra given by Barnes, Greenlees and Kędziorek for finite abelian G. Our calculations show that these spectra are unique as naive-commutative ring spectra in the sense that they are determined up to weak equivalence by their homotopy groups. We further deduce a structure theorem for module spectra over rational equivariant complex K-theory.

Supersymmetric Euclidean field theories and K-theory

We construct spaces of 1-dimensional supersymmetric Euclidean field theories and show that they represent real or complex K-theory. This is done without prescribing the space of states in the definition of field theory.

Higher topological Hochschild homology of periodic complex K-theory

We describe the topological Hochschild homology of the periodic complex K-theory spectrum, THH(KU), as a commutative KU-algebra: it is equivalent to KU[K(Z,3)] and to F(ΣKUQ), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F(ΣKUQ)≃KU∨ΣKUQ, a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element.Then, we prove that THHn(KU), the n-fold iteration of THH(KU), i.e. Tn⊗KU, is equivalent to KU[G] where G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU-algebra on a rational KU-module. We prove that Sn⊗KU is equivalent to KU[K(Z,n+2)] and to F(ΣnKUQ). We describe the topological André-Quillen homology of KU as KUQ.

REAL K-COHOMOLOGY OF COMPLEX PROJECTIVE SPACES

In this paper, we determine the structure of KO-cohomology of complex projective space P l and its product space P l × P m as algebras over the coefficient ring KO ∗ . We also give a description of the map KO ∗ ( P l+m ) → KO ∗ ( P l × P m ) induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product KO ∗ ( P l ) ⊗KO∗ KO ∗ ( P m ) → KO ∗ ( P l × P m ) to see that non-existence of the formal group structure on KO ∗ ( P ∞ ). Introduction A commutative ring spectrum E is said to be complex oriented if an element x of the reduced E-cohomology of the infinite dimensional complex projective space CP ∞ is given such that x maps to a generator of the reduced E-cohomology of 1-dimensional complex projective space CP 1 ((2)). We call such an element x a complex orientation of E. On the other hand, if E-homology E∗E of E is a flat over the coefficient ring E∗, E∗E has a structure of a Hopf algebroid and E-homology theory takes values in the category of E∗E-comodule, in other words, the category of representations of the groupoid represented by the affine groupoid scheme represented by E∗E ((1)). If E is a complex oriented ring spectrum, the E-cohomology of the complex projective space is just a truncated polynomial algebra over E∗ and it is shown that E-homology E∗E of E is a flat over E∗. Moreover the product structure of CP ∞ gives a one dimensional formal group law over E ∗ ((5)) which closely relates with the structure of the Hopf algebroid ((2)). The complex K-theory is one of the most basic examples of complex oriented cohomology theories. However, KO-spectrum representing the real K-theory is one of a few well-known examples of spectra E without any complex orientation such that E∗E is flat over E∗ ((2), (7)). In fact, we see that KO-spectrum does not have any complex orientation by showing that the Atiyah-Hirzebruch spectral sequence converging to KO ∗ (CP l ) has a non-trivial

Coarse Geometry and Topological Phases

We propose the Roe C*-algebra from coarse geometry as a model for topological phases of disordered materials. We explain the robustness of this C*-algebra and formulate the bulk-edge correspondence in this framework. We describe the map from the K-theory of the group C*-algebra of Z^d to the K-theory of the Roe C*-algebra, both for real and complex K-theory.

Towards an understanding of ramified extensions of structured ring spectra

AbstractWe propose topological Hochschild homology as a tool for measuring ramification of maps of structured ring spectra. We determine second order topological Hochschild homology of the p-local integers. For the tamely ramified extension of the map from the connective Adams summand to p-local complex topological K-theory we determine the relative topological Hochschild homology and show that it detects the tame ramification of this extension. We show that the complexification map from connective topological real to complex K-theory shows features of a wildly ramified extension. We also determine relative topological Hochschild homology for some quotient maps with commutative quotients.

Spectral sequences in smooth generalized cohomology

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic study of torsion in differential cohomology. We apply this in detail to smooth Deligne cohomology, differential topological complex K-theory, and to a smooth extension of integral Morava K-theory that we introduce. In each case we explicitly identify the differentials in the corresponding spectral sequences, which exhibit an interesting and systematic interplay between (refinement of) classical cohomology operations, operations involving differential forms, and operations on cohomology with U(1) coefficients.

Notes on the Chern-Character

Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the note a proof of the existence of a Chern-character from complex K-theory to any cohomology Lie theory with values in graded Q-algebras equipped with a theory of characteristic classes is given. It respects the Adams and Steenrod operations.

Nilpotence and descent in equivariant stable homotopy theory

Let G be a finite group and let F be a family of subgroups of G. We introduce a class of G-equivariant spectra that we call F-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of F-nilpotent G-spectra, which are explored further in the sequel to this paper.In the rest of the paper, we prove several general structure theorems for ∞-categories of module spectra over objects such as equivariant real and complex K-theory and Borel-equivariant MU. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex K-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.

Homotopy colimits of classifying spaces of abelian subgroups of a finite group

The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial $2$-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups, $B(2,G)$ does not have the homotopy type of a $K(\pi,1)$ space. For a finite group $G$, we compute the complex K-theory of $B(2,G)$ modulo torsion.

Weighted projective spaces and iterated Thom spaces

For any (n+1)-dimensional weight vector chi of positive integers, the weighted projective space P(chi) is a projective toric variety, and has orbifold singularities in every case other than CP^n. We study the algebraic topology of P(chi), paying particular attention to its localisation at individual primes p. We identify certain p-primary weight vectors pi for which P(pi) is homeomorphic to an iterated Thom space over S^2, and discuss how any P(chi) may be reconstructed from its p-primary factors. We express Kawasaki's computations of the integral cohomology ring H^*(P(\chi);Z) in terms of iterated Thom isomorphisms, and recover Al Amrani's extension to complex K-theory. Our methods generalise to arbitrary complex oriented cohomology algebras E^*(P(chi)) and their dual homology coalgebras E_*(P(\chi)), as we demonstrate for complex cobordism theory, which is the universal example. In particular, we describe a fundamental class in Omega_{2n}^U(P(\chi)), which may be interpreted as a resolution of singularities.

Central Cohomology Operations and K-Theory

AbstractFor stable degree 0 operations, and also for additive unstable operations of bidegree (0, 0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP〈n⌰ for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.

BIEQUIVARIANT MAPS ON SPHERES AND TOPOLOGICAL COMPLEXITY OF LENS SPACES

Weighted cup-length calculations in singular cohomology led Farber and Grant in 2008 to general lower bounds for the topological complexity of lens spaces. We replace singular cohomology by connective complex K-theory, and weighted cup-length arguments by considerations with biequivariant maps on spheres to improve on Farber–Grant's bounds by arbitrarily large amounts. Our calculations are based on the identification of key elements conjectured to generate the annihilator ideal of the toral bottom class in the ku-homology of the classifying space for a rank-2 abelian 2-group.

Chern classes on differentialK-theory

In this note we give a simple, model-independent construction of Chern classes as natural transformations from differential complex K-theory to differential integral cohomology. We verify the expected behaviour of these Chern classes with respect to sums and suspension.

Augmented Γ-spaces, the stable rank filtration, and a bu analogue of the Whitehead conjecture

We explore connections between (4), where we constructed that interpolate between bu and HZ, and earlier work of Kuhn and Priddy on the Whitehead conjecture and of Rognes on the stable rank filtration in algebraic K-theory. We construct a complex of spectra that is a of an auxiliary complex used by Kuhn- Priddy; we conjecture that this chain complex is exact; and we give some supporting evidence. We tie this to work of Rognes by showing that our auxiliary complex can be constructed in terms of the stable rank filtration. As a by-product, we verify for the case of topological complex K-theory a conjecture made by Rognes about the connectiv- ity (for certain rings) of the filtration subquotients of the stable rank filtration of algebraic K-theory. In (4), we introduced a sequence of spectra{Am} interpolating between the connective complex K-theory spectrum bu and the integral Eilenberg- MacLane spectrum HZ. These new resulted from a general con- struction on permutative categories endowed with an augmentation. In the current work, we explore connections of that construction to other set- tings, in particular to work of Kuhn and Priddy (9) on the Whitehead Con- jecture, and to work of Rognes (15) on the stable rank filtration of algebraic K-theory. Connections to Kuhn and Priddy's work were suggested by the many properties that the Am share with the symmetric powers of the sphere spectrum, Sp m (S), which can also be given as an example of the categorical construction of (4). This led us to call Am the bu-analogue of Sp m (S) and to propose a of the Whitehead Conjecture. On the other hand, our construction was also reminiscent of the stable rank filtration in algebraic K-theory, and this made it natural to ask the exact relationship between the two filtrations for bu. Curiously, the two threads converged: in this paper we construct a of an auxiliary complex 2000 Mathematics Subject Classification. 2010 MSC: Primary 55P48, 19L41 ; Sec- ondary 55P91, 55R45.

Yang–Mills theory over surfaces and the Atiyah–Segal theorem

In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(\Gamma) of a compact Lie group $\Gamma$ to the complex K-theory of the classifying space $B\Gamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson's deformation $K$--theory spectrum $\K (\Gamma)$ (the homotopy-theoretical analogue of $R(\Gamma)$). Our main theorem provides an isomorphism in homotopy $\K_*(\pi_1 \Sigma)\isom K^{-*}(\Sigma)$ for all compact, aspherical surfaces $\Sigma$ and all $*>0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.

On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P1-spectrum MGL of Voevodsky is considered as a commutative P1-ring spectrum. Setting \(\mathrm{MGL}^i = \bigoplus_{p-2q =i}\mathrm{MGL}^{p,q}\) we regard the bigraded theory MGLp,q as just a graded theory. There is a unique ring morphism \(\phi\colon\mathrm{MGL}^0(k)\to\mathbb{Z}\) which sends the class [X]MGL of a smooth projective k-variety X to the Euler characteristic \(\chi(X, \mathcal{O}_X)\) of the structure sheaf \(\mathcal{O}_X\). Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories $$\varphi\colon\mathrm{MGL}^{\ast}(X,X-Z) \otimes_{\mathrm{MGL}^{0}(k)} \mathbb{Z} \xrightarrow{\cong} \mathrm{K}_{- *}(X on Z)= \mathrm{K}^{\prime}_{- *}(Z) $$ on the category \(\mathcal{S}m\mathcal{O}p/S\) in the sense of [6], where K*(XonZ) is Thomason–Trobaugh K-theory and K′* is Quillen’s K′-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented in the sense of [6] and ϕ respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism [1].

The invariant subalgebra and anti-invariant submodule of K*K(p)

AbstractBases are constructed for the homogeneous components of the invariant subalgebra and anti-invariant submodule of the algebra K*K(p) of cooperations for p-local complex K-theory.

Rational computations of the topological K-theory of classifying spaces of discrete groups

We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.