Complex Oriented Cohomology Theories Research Articles

Gromov-Witten Invariants in Complex and Morava-Local K-Theories

AbstractGiven a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum $\mathbb{K}$-theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.

BP-cohomology of projective Stiefel manifolds

In this paper, we compute the $BP$-cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and $BP$-operations to prove new results about equivariant maps between Stiefel manifolds.

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

Complex-projective and lens product spaces

Let \(t\) be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal \(S^1\)-action, and of the \(t\)-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the \(t\)th roots of unity. For a commutative complex-oriented cohomology theory \(h^*\), we determine the \(h^*\)-cohomology ring of these spaces (in terms of the \(t\)-series for \(h^*\), in the case of \(t\)-torsion lens product spaces). When \(h^*\) is singular cohomology with mod 2 coefficients, we also determine the action of the Steenrod algebra. We show that these spaces break apart after a suspension as a wedge of desuspensions of usual stunted complex-projective (\(t\)-torsion lens) spaces. We estimate the category and topological complexity of complex-projective and lens product spaces, showing in particular that these invariants are usually much lower than predicted by the usual dimensional bounds. We extend Davis’ analysis of manifold properties such as immersion dimension, (stable) span, and (stable) parallelizability of real-projective product spaces to the complex-projective and lens product cases.

Transchromatic generalized character maps

In "Generalized Group Characters and Complex Oriented Cohomology Theories", Hopkins, Kuhn, and Ravenel develop a way to study cohomology rings of the form E^*(BG) in terms of a character map. The character map can be interpreted as a map of cohomology theories beginning with a height n cohomology theory E and landing in a height 0 cohomology theory with a rational algebra of coefficients that is constructed out of E. In this paper we use the language of p-divisible groups to extend their construction for Morava E-theory so that the character map can land in every height t between 0 and n.

Generalized Artin and Brauer induction for compact Lie groups

Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K G , and to the Borel cohomology associated with any complex oriented cohomology theory. The coefficient ring of K G is the representation ring R(G) of G. When G is a finite group the induction theorems for K G coincide with the classical Artin and Brauer induction theorems for R(G).

Representation and character theory in 2-categories

We develop a (2-)categorical generalization of the theory of group representations and characters. We categorify the concept of the trace of a linear transformation, associating to any endofunctor of any small category a set called its categorical trace. In a linear situation, the categorical trace is a vector space and we associate to any two commuting self-equivalences a number called their joint trace. For a group acting on a linear category V we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors. We call this function the 2-character of V . Such functions of commuting pairs (and more generally, n-tuples) appear in the work of Hopkins, Kuhn and Ravenel [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553–594 (electronic)] on equivariant Morava E-theories. We define the concept of induced categorical representation and show that the corresponding 2-character is given by the same formula as was obtained in [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553–594 (electronic)] for the transfer map in the second equivariant Morava E-theory.

T-spectra and Poincaré duality

Frank Adams introduced the notion of a complex oriented cohomology theory represented by a commutative ring spectrum and proved the Poincaré Duality Theorem for this general case. In the current paper we consider oriented cohomology theories on algebraic varieties represented by symmetric commutative ring T -spectra and prove the Duality Theorem, which mimics the result of Adams. This result is held, in particular, for Motivic Cohomology and Algebraic Cobordism of Voevodsky.

Diagrams indexed by Grothendieck constructions

Let I be a small indexing category, G: I op ! Cat be a functor and BG 2 Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (resp. groupoids) on a stack M and presheaves of simplicial sets (resp. groupoids) over M. The motivation for this paper was the study of homotopy theory of (pre)sheaves on a stack. Since the site associated to a stack M is a Grothedieck construction this led us to an investigation of the homotopy theory of diagrams indexed on a category which is itself a Grothendieck construction (of a diagram of small categories). The body of the paper is concerned with analyzing various Quillen pairs between diagram categories. These adjunctions are of general interest and we present some examples not related to the theory of stacks. We conclude the paper with the applications to stacks. Stacks were introduced in algebraic geometry in order to parametrize families of objects when the presence of automorphisms prevented representability by a scheme or even a sheaf [A, DM, Gi]. Recently stacks have come to play an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and in certain situations, conversely, stacks over the moduli stack of formal groups give rise to spectra [G, R2, GHMR, B]. One fundamental example is the spectrum of topological modular forms [Hp] which is associated to the moduli stack of elliptic curves. Classically, stacks were defined as categories fibered in groupoids over a site C which satisfy descent [DM, Definition 4.1]. In [H] we show that a category fibered in groupoids F over C is a stack if and only if the assignment satisfies the homotopy sheaf

Descent for quasi-coherent sheaves on stacks

Stacks were introduced in order to parametrize problems in algebraic geometry where the presence of automorphisms prevented representability by a scheme or even a sheaf (see Artin [1], Deligne–Mumford [3] and Giraud [5]). One early application was Deligne and Mumford’s use of stacks to prove the irreducibility of the space of curves of a given genus [3]. More recently stacks have also played an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and, in certain situations, stacks over the moduli stack of formal groups give rise to spectra (see Goerss [6], Goerss–Hopkins [8] and Rezk [21]) which play an important role in understanding the homotopy groups of spheres (see Goerss–Henn–Mahowald–Rezk [7] and Behrens [2]). One fundamental example is the spectrum of topological modular forms (see Hopkins [12]) which is associated to the moduli stack of elliptic curves.

Computation of generalized equivariant cohomologies of Kac–Moody flag varieties

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain projective varieties X equipped with an algebraic action of a complex torus T, the equivariant cohomology ring H T * ( X ) can be described by combinatorial data obtained from its orbit decomposition. In this paper, we generalize their theorem in three different ways. First, our group G need not be a torus. Second, our space X is an equivariant stratified space, along with some additional hypotheses on the attaching maps. Third, and most important, we allow for generalized equivariant cohomology theories E G * instead of H T * . For these spaces, we give a combinatorial description of E G * ( X ) as a subring of ∏ E G * ( F i ) , where the F i are certain invariant subspaces of X. Our main examples are the flag varieties G / P of Kac–Moody groups G , with the action of the torus of G . In this context, the F i are the T-fixed points and E G * is a T-equivariant complex oriented cohomology theory, such as H T * , K T * or MU T * . We detail several explicit examples.

An Equivariant Analog of the Poincaré―Hopf Theorem

A new method for localization of algebro-topological invariants of smooth manifolds via equivariant tangent vector fields is presented. Main realizations of direct image constructions (the Gysin map and the Becker―Gottlieb transfer map) are calculated for Grassmannizations of complex vector bundles and for a complex-oriented cohomology theory. Bibliography: 12 titles.

Chromatic characteristic classes in ordinary group cohomology

We study a family of subrings, indexed by the natural numbers, of the mod p cohomology of a finite group G. These subrings are based on a family of v n -periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of G, naturally interpolating between the work of Quillen on var(H ∗(BG)) , the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, var( Ch( G)). Our subrings give rise to a ‘chromatic’ (co)filtration, which has both topological and algebraic definitions, of var(H ∗(BG)) whose final quotient is the variety var( Ch( G)).

Realization of Abel's universal formal group law

To a complex oriented cohomology theory $h^*(-)$ one may assign a formal group law $F_h(u,v)$ over $h^*(pt)$ . The purpose of this paper is to show that the converse holds true for the case of Abel's universal formal group law ${\cal F}_{Ab}(u,v)$ , i.e. we will prove the existence of a complex oriented cohomology theory $Ab^*(-)$ whose formal group law $F_{Ab}$ is isomorphic to ${\cal F}_{Ab}(u,v)$ .

Equivariant formal group laws and complex oriented cohomology theories

The article gives an introduction to equivariant formal group laws, and explains its relevance to complex oriented cohomology theories in general and to complex cobordism in particular.

Equivariant orientation theory

The article gives an introduction to equivariant formal group laws, and explains its relevance to complex oriented cohomology theories in general and to complex cobordism in particular.

Generalized group characters and complex oriented cohomology theories

Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex.

Operations and quantum doubles in complex oriented cohomology theory

We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the LandweberNovikov algebra, as represented by a subalgebra of operations in double complex cobordism. We work in the context of Boardman’s eightfold way, which offers an important framework for clarifying the relationship between quantum doubles and the standard machinery of Hopf algebroids of homology cooperations. These considerations give rise to novel structures in double cohomology theory, and we explore the twist operation and extensions of the quantum antipode by way of example.

The complex oriented cohomology of extended powers

We examine the behaviour of a complex oriented cohomology theory G * (-) on D p (X), the C p -extended power of a space X, seeking a description of G * (D p (X)) in terms of the cohomology G * (X). We give descriptions for the particular cases of Morava K-theory K(n) for any space X and for complex cobordism MU, the Brown-Peterson theories BP and any Landweber exact theory for a wide class of spaces.

The structure of the Hopf algebroid associated with the elliptic homology theory

In representation theory of groups, there is an equivalence between the category of affine group and the category of commutative Hopf algebras. This fact can be generalized to affine groupoid schemes, that is, a groupoid scheme is an internal groupoid in the category of and there is an equivalence between the category of affine groupoid and the category of Hopf algebroids. On the other hand, by a work of J.F. Adams on generalized homology theory ([l], [2].) a commutative ring spectrum E such that E*E is flat over E* = E*(S°) gives a Hopf algebroid (£*, E*E). One of the most important examples of such E is the complex cobordism spectrum MU. A theorem of Quillen enable us to identify the affine groupoid scheme represented by the Hopf algebroid defined from the complex cobordism theory with an affine groupoid scheme of formal groups and strict isomorphisms between them. Many homology theories related with complex cobordism (for example, fiP-theory, Morava .fί-theory, elliptic homology) give closed (or locally closed) subgroupoid schemes of above groupoid scheme. This viewpoint is originally due to Jack Morava ([lθ]). We give basic definitions and constructions on internal groupoids in sections 1 and 2. In section 3, we translate the language of previous sections in terms of Hopf algebroids. We give a general description of Hopf algebroids associated with complex oriented cohomology theories satisfying certain condition in Section 4. Finally, we determine the structure of the Hopf algebroid associated with the elliptic homology theory given by Landweber ([8]).