Some results in generalized homology, <i>K</i>-theory and bordism

Ray, Nigel

doi:10.1017/s0305004100050520

Cited by 17 publications

(13 citation statements)

References 16 publications

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“…The only unknown in (2.6) is MSp*, since MSU*(MSp) is a free MSU*-module: it is this spectral sequence that we investigate thoroughly in [3]. Note that the corresponding spectral sequence for ko does not converge, since H*(KO; MSp*) is non-zero in negative dimensions.…”

Section: Algebraic Methodsmentioning

confidence: 99%

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A Note on the Symplectic Bordism Ring

Ray¹

1971

Bulletin of the London Mathematical Society

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Section: Algebraic Methodsmentioning

confidence: 99%

“…The purpose of this note is to announce the author's computation of the symplectic bordism ring MSp* in dimensions < 20. The details will appear in two subsequent papers [3,4] and here we simply list the methods used and the results obtained.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Symplectic Bordism Ring

Ray¹

1971

Bulletin of the London Mathematical Society

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“…By definition, this is the exponential series of F , and we wish to interpret it geometrically using a convenient variant of a method pioneered by Adams [1]. Given any spectrum F , the Boardman homomorphism (as expounded in [7], for example) may be applied to the rationalisation F Q to yield an isomorphism…”

Section: The Rational Relationsmentioning

confidence: 99%

A rational approach to Hopf rings

Hunton

Ray

1995

Journal of Pure and Applied Algebra

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We study Hopf rings of the form E * (F *), where E * (−) is a complex oriented homology theory and F * is the graded ring space arising from a complex oriented Ω-spectrum F. We propose a new model Hopf ring E Q * (F *) which is constructed by purely algebraic means from the rationalisations EQ * (F *), E * (F Q *) and EQ * (F Q *), and which in many cases is isomorphic to E * (F *). Our model may be expressed in terms of the conjugate Bell polynomials associated with the exponential series for E and F , and we explain how calculations may be carried out in this context. We discuss the relationship between E Q * (F *) and the well-known model E R * (F *) due to Ravenel and Wilson.

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“…The analytic and function theoretic viewpoint has prospered in Russian literature, whereas the homotopical approach has featured mainly in the west; the machinery of formal group laws, however, has been well-oiled by both schools, and the rôle of index theory has been enhanced by many authors since it was originally suggested by Gelfand [40]. Ironically, the dichotomy between the first and second is illustrated by comparing computations of the Chern-Dold character [15] with applications of the Boardman homomorphism and Hurewicz genus [79]. In essence, these bear witness to the remarkable influence of Novikov and Adams respectively.…”

Section: Introductionmentioning

confidence: 99%

Toric Genera

Buchstaber

Panov

Ray

2010

International Mathematics Research Notices

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Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T k . In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus Φ, which was introduced independently by Krichever and Löffler in 1974, albeit from radically different viewpoints. In fact Φ is a version of tom Dieck's bundling transformation of 1970, defined on T k -equivariant complex cobordism classes and taking values in the complex cobordism algebra Ω * U (BT k + ) of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera, and refer to the index theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework, and to introduce parametrised versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever's generalised elliptic genus as universal amongst genera that are rigid on SU -manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.

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Some results in generalized homology, K-theory and bordism

Cited by 17 publications

References 16 publications

A Note on the Symplectic Bordism Ring

A Note on the Symplectic Bordism Ring

A rational approach to Hopf rings

Toric Genera

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