Cohomology Operations Research Articles

Evaluating a p-th order cohomology operation

A certain p-th order cup product is detected by a p-th order cohomology operation. The result is applied to finite H-spaces, to show that several properties of compact Lie groups do not hold for arbitrary torsion free finite H-spaces.

A note on immersing manifolds in euclidean spaces

Let M be a closed, connected smooth and 3-connected mod 2 (that is Hi(M;ℤ2) = 0, 0 < i ≤ 3) manifold of dimension n = 7 + 8k. Using a combination of cohomology operations on certain cohomology classes of M and on the Thom class of the stable normal bundle of M we show that under certain conditions M immerses in R2n−8. This extends previously known results for such a general manifold when the number of 1's in the dyadic expansion of n is less than 8.

Derivation of the gauge-invariant action for open and closed free bosonic string field theories.

The gauge-invariant actions for open and closed free bosonic string field theories are obtained from the string field equations in the conformal gauge using the cohomology operations of Banks and Peskin. For the closed-string theory no restrictions are imposed on the gauge parameters.

BRS cohomology and Casimir operators

Casimir operators play a central role in the study of cohomology problems for semisimple Lie algebras. An attempt is made to generalize this to strings. This generalization may be useful for studying small oscillations around nontrivial backgrounds.

The Relation Between Stable Operations for Connective and Non-Connective p-Local Complex K-Theory

AbstractThe question of which degree 0 stable cohomology operations for connective K-theory localized at a prime p arise from operations for non-connective K-theory is investigated. A necessary and sufficient condition is established, and an example of a connective operation not arising in this way is constructed.

Frame Fields on Manifolds

Consider the following stable secondary cohomology operations associated with the relations in the mod 2 Steenrod algebra: such thatLet ψ5 be a stable tertiary cohomology operation associated with the above relation. We assume that (ϕ4, ϕ5) and ψ5 are chosen to be spin trivial in the sense of Theorem 3.7 of [14].Let ϕ0,0, ϕ1,1 be the stable Adams basic secondary cohomology operations associated with the relations:respectively.

Pyramids of higher order cohomology operations in the $p$-torsion-free category.

Pyramids of higher order cohomology operations in the $p$-torsion-free category.

Product structures in pyramids of higher order cohomology operations.

Product structures in pyramids of higher order cohomology operations.

Operations in Segal's cohomology

Operations in Segal's cohomology

Secondary cohomology operations that detect homotopy classes

Secondary cohomology operations that detect homotopy classes

There are no phantom cohomology operations inK-theory

There are no phantom cohomology operations inK-theory

Homology operations and power series

Bullett and Macdonald [1] have used power series to simplify the statement and proof of the Adem relations for Steenrod cohomology operations. In this paper I give a similar treatment of May's generalized Adem relations [4, §4] and of the Nishida relations ([6], [2, 1.1.1(9)], [5, 3.1(7)]). Both sets of relations apply to Dyer-Lashof operations in E∞, spaces such as infinite loop spaces ([3], [2, I.I]) and in H^ ring spectra ([5, §3]).

Cohomology operations and $H$-spaces

Cohomology operations and $H$-spaces

Higher order cohomology operations in the p-torsion-free category

Algebraic systems of higher order cohomology operations derived from the p-divisibility of the Chern character are considered here. Two such systems, dual to one another, are constructed and several applications thereof are given. Higher order product formulae are also considered.

A Formula for Deviation from Commutativity: The Transfer and Steenrod Squares

The ordinary cohomology transfer associated to the orbit space projection of a finite group action need not commute with stable cohomology operations. In particular, if an even group acts on a space, the resulting transfer $\tau$ will not generally commute with the Steenrod squares, ${\text {S}}{{\text {q}}^i}$. This paper contains a formula for the deviation from commutativity $({\text {S}}{{\text {q}}^i}\tau - \tau {\text {S}}{{\text {q}}^i})x$ in the case of an involution. The formula involves the restriction of $x$ to the cohomology of the fixed point set, as well as certain naturally occurring characteristic classes.

A formula for deviation from commutativity: the transfer and Steenrod squares

The ordinary cohomology transfer associated to the orbit space projection of a finite group action need not commute with stable cohomology operations. In particular, if an even group acts on a space, the resulting transfer τ \tau will not generally commute with the Steenrod squares, S q i {\text {S}}{{\text {q}}^i} . This paper contains a formula for the deviation from commutativity ( S q i τ − τ S q i ) x ({\text {S}}{{\text {q}}^i}\tau - \tau {\text {S}}{{\text {q}}^i})x in the case of an involution. The formula involves the restriction of x x to the cohomology of the fixed point set, as well as certain naturally occurring characteristic classes.

Cohomology operations in a category

The purpose of this paper is to make clear that the concepts of cohomology and cohomology operations, as well as homotopy operations, are very natural ones and in fact arise with respect to any object or set of objects in a category which has enough structure for the definitions to make sense. This means, for example, that given an object A in a category .dwe can use the object A to form a sequence of algebraic “pictures” or algebraic higher order structure of the category :/taken relative to A. We confine ourselves here to the general construction of primary and secondary operations, carried out in the first two sections. It should be noted, however, that this is

Γ-SPACES, ORIENTATIONS AND COHOMOLOGY OPERATIONS

Journal Article Γ-SPACES, ORIENTATIONS AND COHOMOLOGY OPERATIONS Get access RICHARD WOOLFSON RICHARD WOOLFSON University of ManchesterManchester 13 Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 31, Issue 3, September 1980, Pages 363–383, https://doi.org/10.1093/qmath/31.3.363 Published: 01 September 1980 Article history Received: 05 July 1978 Revision received: 20 October 1979 Published: 01 September 1980

Cohomology operations in the loop space of the compact exceptional group F

Cohomology operations in the loop space of the compact exceptional group F

Homotopy classification of connected sums of sphere bundles over spheres, I

Let Biy i —1,2, • • • , & , be ^-sphere bundles over g-spheres (p, It is understood that each Bt also denotes the total space of the bundle and has the oriented differentiable structure induced from those of the fibre and the base space. We denote the connected sum of Biy i = I, 2, • • , k, by %%=iBi. The necessary and sufficient conditions for such two connected sums of sphere bundles over spheres to be homotopy equivalent were given in [5] and [6]. In [5], we treated with the case that every bundle admits a cross-section, and in [6], we discussed the general case. As special cases of [5], in the preceding paper [22], we classified the connected sums of ^-sphere bundles over g-spheres which admit crosssections for (p, q) = (n, rc + 1) (n^2) , (n — 1, ra + 1) (?z^4), and (n — 2, 7Z + 1) (w^>6), and the results were applied to classify certain manifolds with sufficient connectedness. In this paper, by applying [6], we completely classify the connected sums of ^-sphere bundles over ^-spheres which do not necessarily have cross-sections up to homotopy equivalence for (p9q) = (n-lyn + l) 4) and (n-2,n + l) (rcS>6). Let (p,q) = (n-I,n + I) (n^4) or (n-2, n + V) (^6). A connected sum fjLiBi is called of type O if each Bt admits a cross-section, of type I if any Bt admits no cross-section, and of type (O-j-I) if there exist Bi admitting a cross-section and Bj admiting no cross-section. These definitions coincide with those defined in [3] and [4] using S% and Adem's secondary cohomology operation. Hence, the types are homotopically invariant. Furthermore, if s bundles admit cross-sections and t bundles