Cohomology Operations Research Articles

Cohomology Operations and the Hopf Algebra Structures of the Compact, Exceptional Lie Groups E 7 and E 8

Cohomology Operations and the Hopf Algebra Structures of the Compact, Exceptional Lie Groups <i>E</i> ₇ and <i>E</i> ₈

Hopf algebra structure and cohomology operations of the mod 2 cohomology of the exceptional Lie gronps

Hopf algebra structure and cohomology operations of the mod 2 cohomology of the exceptional Lie gronps

Generalized higher order cohomology operations and stable homotopy groups of spheres

Generalized higher order cohomology operations and stable homotopy groups of spheres

A universal isomorphism for p-typical formal groups and operations in Brown-Peterson cohomology

We construct an abstract isomorphism of p-typical formal groups which is universal for isomorphisms of p-typical formal groups over Z (p) -algebras or characteristic zero rings. Associated to this universal isomorphism is a homomorphism of rings Z[V 1, V 2, …] → Z[V 1, V 2, …; T 1, T 2, …] which (after localization at p) can be identified with the right unit homomorphism η R : BP ∗( pt) → BP ∗( BP) of the Hopf-algebra BP ∗( BP) of Brown-Peterson (co)homology. We calculate η R modulo the ideal ( T 1, T 2, …) 2. These results are then used to obtain information on some of the operations of Brown-Peterson cohomology.

A classification of secondary cohomology operations

A classification of secondary cohomology operations

A Relation Between Obstructions and Functional Cohomology Operations

A Relation Between Obstructions and Functional Cohomology Operations

A relation between obstructions and functional cohomology operations

In a recent paper, P. Olum developed a formula which relates the obstruction to an extension and the induced homomorphism in cohomology. In the present paper, we develop an analogous formula which relates the obstruction to an extension and a certain functional cohomology operation. This operation is a generalization of the standard functional cohomology operation of Steenrod. Olum showed that his formula is particularly useful when the range space of the extension problem is one of the classifying spaces BU, BO, or BSp. Our formula is also useful in this context and we show that it can be used to compute the obstruction in certain situations where the Olum formula fails.

The Products in the Steenrod Rings of the Complex and Sympletic Cobordism Theories

This paper is concerned with the Hopf algebra structure of a certain subalgebra S of the Steenrod ring AG of stable cohomology operations in the complex cobordism theory and the sympletic cobordism theory MG*( ), where G=U or Sp, the infinite dimensional unitary or sympletic group, respectively. The Hopf algebra structure and its applications of Steenrod algebra with coefficients Zp, for a prime p, have long been studied by many topologists (Steenrod-Epstein [6]). Novikov [3] investigated the Steenrod rings of generalized cohomology theories. Landweber [1] also studied general properties of AG as Hopf algebra. The main purpose of this paper is to determine the explicit product formula in S (Theorem 3.1) and the indecomposable quotient S/S2 (Theorem 4.1), where S denotes the kernel of the augmentation S-»Z. We use the following notations. Let Z be the ring of integers and Zm = Z/mZ. According to Landweber [1], AG can be expressed as AG = A®S, with the coefficient A = Q^MG*(point). In case G=£7, /t=Z[xl5 x2,...], deg(Xj)=-2/. In case G = Sp,A has not been determined completely. The subalgebra S is a Hopf algebra over Z and has a Z-free basis {Sj}, with dQg(SI) = d^rrir9 where d=2 or 4 according as G=17 or Sp and I=(il9 i2,...) is a sequence of non-negative integers such that all but a finite number of ir are zero. For two Z-graded modules M=^i^L_aoMi and N^^^-^ Ni9 the completed tensor product

An inversion formula and cohomology operations

An inversion formula and cohomology operations

Generalised Cohomology Operations and the Flat Product

Generalised Cohomology Operations and the Flat Product

The Todd character and cohomology operations

In “On the Conflict of Bordism of Finite Complexes” [ J. Differential Geometry], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms th r: MU s(X) → H s→r(X; Q) . In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc. 57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.

Annihilator ideals and primitive elements in complex bordism

We determine the prime ideals in the complex bordism ring $\pi_{\ast}(MU)$ which can be the annihilator ideals of elements in the complex bordism of finite complexes. Such a prime ideal must be finitely generated and invariant under all stable $MU$-cohomology operations. We go on to determine the invariant prime ideals in $\pi_{\ast}(MU)$.

A definition of exotic characteristic classes of spherical fibrations

The object of this paper is to define certain characteristic cohomology classes for spherical fibrations which are zero on vector bundles and to show that the classes defined are not always zero. For an introductory survey of characteristic classes and spherical fibrations the reader is referred to [12] and the references therein. Briefly there is a 'structure group' G and a classifying space BG for spherical fibrations and similar spaces (SG and BSG) for oriented spherical fibrations. SG has the homotopy type of lim,_~o~ (f2S)1, where (f2S)1 is the space of degree 1 basepoint preserving maps of to itself. The cohomology of all four spaces has recently been computed by Milgram ([I0]), May ([8]) and Tsuchiya ([18]). My object is to define certain classes e k e H r ~ I ( B S G ; Zp) for p an odd prime (where r = 2 p 2 ) and ekeH2~l l (BG; Z2) which I will refer to as exotic classes. In order to simplify notation I will only deal with the case o fp odd, but all of the theorems herein can be proved for p = 2 with the obvious changes in notation. All cohomology groups will have Zp coefficients unless otherwise indicated. The definition given here is similar to one given by Peterson in [13] and to a definition of e~ given by Gitler-Stasheff in [4]. In each case the exotic classes are defined in terms of twisted secondary cohomology operations (TSCO's) acting on the Thom class u e H * M S G , where MSG is the Thorn space of the universal bundle over BSG. TSCO's were introduced by Thomas ([16]) and axiomatized by McClendon ([9]). They are a generalization of ordinary secondary operations to the category of topological pairs (X, V) over a fixed space Y. The analogue of the Steenrod algebra in this category is A (Y) where A is the Steenrod algebra and A ( Y ) = H * Y| as a vector space with the multiplication appropriate to defining an A (Y) module structure on H* (X, V). TSCO's are derived from relations in A(Y) jus t as ordinary secondary operations are derived from relations in A. Indeed, ordinary secondary operations can be regarded as TSCO's for the special case Y=pt. Now the Thorn space of any oriented spherical fibration can be regarded as a pair over BSG, so relations in A (BSG) could be used to define characteristics classes on suitable spherical fibrations. In [13] Peterson defined an algebra injection O:A-~ A (BSG) with the property that O(a) annihilates the Thom class u of MSG

Cohomology Operations on p-Fold Sums

Cohomology Operations on p-Fold Sums

On Transportable Forms

In the theory of deformations of compact complex manifolds, the hypothesis of constancy of the dimension of diverse structural cohomology groups pertaining to a fibre plays an important role (see, for instance, [3, Propositions 2.5 and 2.7, Theorems 2.2 and 2.3, and Definition 6.1]). This paper is the first of two devoted to the investigation of conditions under which constancy of the dimension of given cohomology groups is assured, and more generally, to the study of the variation of that dimension.In [2] Griffiths introduces extendible forms in a holomorphic deformation. We consider in this paper a differentiate family, and besides extendible, also co-extendible and transportable forms (see §5), and deduce from their existence conclusions about the variation of the dimension of the corresponding structural cohomology groups. It is left to a subsequent paper to give more explicit conditions by means of cohomology operations, and to deal with some applications.

Cohomology operations on 𝑝-fold sums

A formula is given for evaluating higher order cohomology operations on integral classes that are p-fold multiples, p a prime.

On the Homotopy Type of Stiefel Manifolds

Suppose that we have a fibre bundle where the total space has the same homotopy type as the product of the fibre and the base. When can we conclude that the bundle is trivial, in the sense of fibre bundle theory? This question arises in the classification theory of Hopf homogeneous spaces, especially in relation to Stiefel manifolds. Results are proved, using cohomology operations, which answer the question in some cases.

Primary cohomology operations in BSJ

Primary cohomology operations in BSJ

Cohomology operations in local coefficient theory

Cohomology operations in local coefficient theory

On some secondary cohomology operations

On some secondary cohomology operations