Complex Cobordism Research Articles

Symmetric Functions, Formal Group Laws, and Lazard's Theorem

Lazard's theorem is a central result in formal group theory; it states that the ring over which the universal formal group law is defined (known as the Lazard ring) is a polynomial algebra over the integers with infinitely many generators. This ring also shows up in algebraic topology as the complex cobordism ring. The main aim of this paper is to show that the polynomial structure of the Lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients. The connection between symmetric functions and the Lazard ring is provided by a certain Hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law. We study this map by deriving formulas for the images of certain symmetric functions; in passing, we use this map to prove some symmetric function and Catalan number identities. Based on the above results, we prove Lazard's theorem, and present an application to the construction of certainp-typical formal group laws over the integers. Combinatorial methods play a major role throughout this paper.

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.

Localization and Completion Theorems for MU-Module Spectra

Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU ; examples include all of the standard homotopical MU -modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M∗(BG) and M∗(BG). The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG, and there is an MUG-module MG whose underlying MU -module is M . This allows the use of topological analogues of constructions in commutative algebra. The computation ofM∗(BG) andM∗(BG) is expressed in terms of spectral sequences whose respective E2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU ∗ and its module M ∗ . The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global I∗-functor with smash product.

Torsion algebraic cycles and complex cobordism

Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integral coefficients [3]. That conjecture predicted that every integral cohomology class of Hodge type (p, p) on a smooth projective variety should be the class of an algebraic cycle, but Atiyah and Hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle. Here we provide a more systematic explanation for their results by showing that the classical cycle map, from algebraic cycles modulo algebraic equivalence to integral cohomology, factors naturally through a topologically defined ring which is richer than integral cohomology. The new ring is based on complex cobordism, a well-developed topological theory which has been used only rarely in algebraic geometry since Hirzebruch used it to prove the Riemann-Roch theorem [17]. This factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by Atiyah and Hirzebruch. It goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective, as well as nonsurjective. The kernel of the classical cycle map is called the Griffiths group, and the topological proof here that the Griffiths group can be nonzero is the first proof of this fact which does not use Hodge theory. (The proof here gives nonzero torsion elements in the Griffiths group, whereas Griffiths's Hodge-theoretic proof gives nontorsion elements [13].) This topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before, thus answering some questions posed by Colliot-Thelene and Schoen ([8], p. 14; [37], p. 13). Colliot-Thelene asked, in particular, whether the map CH2(X)/n -* H4(X, Z/n) is injective for all smooth complex projective varieties X. Here CH'X is the group of codimension i algebraic cycles modulo rational equivalence. The first examples where Colliot-Thelene's map is not injective were found by Kollar and van Geemen (see [4], p. 135); very recently, Bloch and Esnault found examples defined over number fields [7]. (Over nonalgebraically closed fields k there are other examples of smooth projective varieties Xk with CH2 (Xk)/n -* Hg4t(Xk, Z/n) not injective, due to Colliot-Thelene and Sansuc as reinterpreted by Salberger (see [9] and [8], Remark 7.6.1), and Parimala and Suresh [31]. These elements of CH2 (Xk)/n are not shown to remain nonzero in CH2(Xc)/n, however.) Here our topological method

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]).

Dyer–Lashof operations in the Hopf ring for complex cobordism

AbstractWe give a complete description of the additive Dyer-Lashof operations on the Hopf ringwith coefficients in. We re-prove a result of Ravenel and Wilson [9] giving the operations on the elementand give formulae for the operations on the other generators.

Schubert Calculus in Complex Cobordism

We study the structure of the complex cobordism ring of the flag variety of a compact connected Lie group. An explicit procedure for determining products of basis elements is obtained, generalizing the work of Bernstein-Gel’fand-Gel’fand on ordinary cohomology and of Kostant-Kumar on $K$-theory. Bott-Samelson resolutions are used to replace the classical basis of Schubert cells.

Schubert calculus in complex cobordism

We study the structure of the complex cobordism ring of the flag variety of a compact connected Lie group. An explicit procedure for determining products of basis elements is obtained, generalizing the work of Bernstein-Gel'fand-Gel'fand on ordinary cohomology and of Kostant-Kumar on K-theory. Bott-Samelson resolutions are used to replace the classical basis of Schubert cells

Complex cobordism ring and conformal field theory over Z

Complex cobordism ring and conformal field theory over Z

On Chern classes of stably fibre homotopic trivial bundles

Let ξ be a stably fibre homotopic trivial vector bundle. A classical result of Thorn states that the Stiefel-Whitney classes of ξ vanish, and one way to prove this is as follows. Letube the Thorn class of ξ in mod 2 cohomology. Thenuis stably spherical by [2] and therefore all stable cohomology operations vanish onu, showing thatwi(ξ)u= Sqiu= 0. In this note we shall apply this same method using complex cobordism and Landweber-Novikov operations to study relations among Chern classes of a stably fibre homotopic trivial complex vector bundle. We will thus obtain in a unified way certain strong modpconditions for every primep.

Book Review: Complex cobordism and stable homotopy groups of spheres

Book Review: Complex cobordism and stable homotopy groups of spheres

The Conner-Floyd Map for Formal A-Modules

A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1 is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

The Complex Cobordism of BO n

The Complex Cobordism of BO_<i>n</i>

Localization of stable homotopy rings

V. P. Snaith has shown that a number of interesting homology theories (e.g. K-theory, complex cobordism) arise when one localizes the stable homotopy type of certain H-spaces (BU(1), BU, etc.). In this paper we sketch a construction which, given an H-space X and , produces a ring spectrum IΣ−nX which lies between Xand its localization with respect to b, and in many cases has a tractable structure. We apply this when X is an Eilenberg-MacLane space K(ℤ/, n) and is a generator (p = 0 or a prime), and get complete results on the localizations in this case. The results confirm one's suspicions that localization is a fairly drastic manoeuvre, cutting down the large and complicated ring to something more or less trivial, except when p = 0 and n = 2, in which case the rich geometry of K(ℤ, 2) = BU(1) gives complex K-theory.

Actions on Invariant Spheres around Isolated Fixed Points of Actions of Cyclic Groups

Fix a prime number p and let Zp be a cyclic group of order p. We consider a pair (M, 0) consisting of a compact simply connected almost complex manifold M without boundary and a smooth Zp-action 0: ZPX M—>M preserving the almost complex structure of M. We suppose that M is given an invariant Riemannian metric. If a(eM) is an isolated fixed point, then the induced action of Zp on the tangent space at a gives a complex Zp-module Va which has no trivial irreducible factor. Let f : EZp—*BZp be a universal principal Zp-bundle and let f (Vtt) : EZpXzp Va —>BZP be the V0-bundle associated with ?. If a and b are isolated fixed points, we compare the cobordism Euler classes 0(f (Va)) and e(£ (Vb)) which belong to the complex cobordism group MU* (BZP) of the classifying space BZP of Zp. Let .PV be the universal formal group law over MU*9 and write

Some Calculations In the Unstable Adams-Novikov Spectral Sequence

The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {Er(X)}9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £2term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S2+1. The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E2-term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S2n+1. We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the

The Morava K-Theories of Eilenberg-MacLane Spaces and the Conner-Floyd Conjecture

Introduction. Of the many generalized homology theories available, very few are computable in practice except for the simplest of spaces. Standard homology and K-theory are the only ones which can be considered somewhat accessible. In recent years, complex cobordism, or equivalently, Brown-Peterson homology, has become a useful tool for algebraic topology. The high state of this development is particularly apparent with regard to BP stable operations, which are understood well enough to have many applications to stable homotopy; see for example [16]. Despite this achievement, it is still virtually impossible to compute the Brown-Peterson homology of any but the nicest of spaces; for example: some simple classifying spaces, spaces with no torsion and spaces with few cells. As a replacement for Brown-Peterson homology in this respect, we present the closely related generalized homologies known as the Morava K-theories. These are a sequence of homology theories, K(n) *(-), n > 0, for each prime p. The n = 1 case is essentially standard mod p complex K-theory. These theories are periodic of period 2(pn - 1) and fit together to give Morava's beautiful structure theorem for complex cobordism; see [11]. Because of their close relationship to complex bordism, information about them will sometimes suffice for bordism, and thus geometric, problems. This is the case with our proof of the Conner-Floyd conjecture. The Morava K-theories each possess Kiunneth isomorphisms for all spaces. This feature enhances their computability tremendously. We demonstrate this point by computing the Morava K-theories of the Eilenberg-MacLane spaces. These spaces are difficult to handle even for

On Adams' splitting of K-theory and complex cobordism

In (2) Adams gave a splitting of complex K-theory with coefficients in the ring R(d) of rationals a/b such that b contains no prime p with p ≡ 1 (mod d). The splitting comes from a complete set of projection operators on K(X; R(d)). One of the operators is then used to obtain a stable, multiplicative idempotent ε on complex cobordism with coefficients in the same ring R(d) and hence a splitting of the representing spectrum MUR(d). However, the idempotent is initially defined over the rational numbers and work is needed to show that it actually gives an operation on MUR(d). Since Novikov (6) has shown that multiplicative cobordism operations are distinguished by their values on the generator ω ∈ MU2CP∞, it is natural to seek an explicit formula for ε(ω) which wi11 show that ε gives an operation over the subring R(d).

Constructing formal groups III: applications to complex cobordism and Brown-Peterson cohomology

In parts I and II, cf. [S], [6] and also [7], of this series of papers we constructed a universal p-typical one dimensional commutative formal group and a universal one dimensional commutative formal group. The extraordinary cohomology theories BP (Brown-Peterson cohomology) and MU (complex cobordism cohomology) are complex oriented and hence define one dimensional formal groups over BPt Z, Tz, . . . ] (when localized at p) can be identified with the right unit map pR : BP,(ptj+ BP,(BP) of the Hopf algebra BP&BP). ln Sections 4, 5 belo-w, we use the universal isomorphism of [S] to obtain a recursive description of the homomorphism qR. This description is useful in the calculation of various BP cohomology operatio s, cf. Sections 6,7 below. To obtain this recursive description of qR we need an isomorphism formula (Section 5 below) which is also useful in the theory of formal groups itself, cf. part III of f8].

The Hopf ring for complex cobordism

In [38] Thom defined the unoriented and oriented cobordism rings, soon generalized to complex cobordism by Milnor [19] and Novikov [22]. These geometric constructions were later shown to give rise to generalized homology and cohomology theories [39] by Atiyah [3]. These theories have received a great deal of attention in recent years. In this paper we offer three new things. First, we obtain unstable homotopy theoretic information from formal group laws. Second, we make essential use of the concept of Hopf rings both in the description of our results and in the proofs. Third, we give a detailed analysis of the homology structure of the (unstable) classifying spaces for complex cobordism, including a completely algebraic construction which contains total information about the unstable complex cobordism operations. Some of our results were announced in [29].