505 research outputs found
Homotopy decompositions and K-theory of Bott towers
We describe Bott towers as sequences of toric manifolds M^k, and identify the
omniorientations which correspond to their original construction as toric
varieties. We show that the suspension of M^k is homotopy equivalent to a wedge
of Thom complexes, and display its complex K-theory as an algebra over the
coefficient ring. We extend the results to KO-theory for several families of
examples, and compute the effects of the realification homomorphism; these
calculations breathe geometric life into Bahri and Bendersky's recent analysis
of the Adams Spectral Sequence. By way of application we investigate stably
complex structures on M^k, identifying those which arise from omniorientations
and those which are almost complex. We conclude with observations on the role
of Bott towers in complex cobordism theory.Comment: 26 page
Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes
We extend work of Davis and Januszkiewicz by considering {\it omnioriented}
toric manifolds, whose canonical codimension-2 submanifolds are independently
oriented. We show that each omniorientation induces a canonical stably complex
structure, which is respected by the torus action and so defines an element of
an equivariant cobordism ring. As an application, we compute the complex
bordism groups and cobordism ring of an arbitrary omnioriented toric manifold.
We consider a family of examples , which are toric manifolds over
products of simplices, and verify that their natural stably complex structure
is induced by an omniorientation. Studying connected sums of products of the
allows us to deduce that every complex cobordism class of dimension
>2 contains a toric manifold, necessarily connected, and so provides a positive
answer to the toric analogue of Hirzebruch's famous question for algebraic
varieties. In previous work, we dealt only with disjoint unions, and ignored
the relationship between the stably complex structure and the action of the
torus. In passing, we introduce a notion of connected sum # for simple
-dimensional polytopes; when is a product of simplices, we describe
P^n# Q^n by applying an appropriate sequence of {\it pruning operators}, or
hyperplane cuts, to .Comment: 22 pages, LaTeX2e, to appear in Internat. Math. Research Notices
(2001
Weighted projective spaces and iterated Thom spaces
For any (n+1)-dimensional weight vector {\chi} of positive integers, the
weighted projective space P(\chi) is a projective toric variety, and has
orbifold singularities in every case other than CP^n. We study the algebraic
topology of P(\chi), paying particular attention to its localisation at
individual primes p. We identify certain p-primary weight vectors {\pi} for
which P(\pi) is homeomorphic to an iterated Thom space over S^2, and discuss
how any P(\chi) may be reconstructed from its p-primary factors. We express
Kawasaki's computations of the integral cohomology ring H^*(P(\chi);Z) in terms
of iterated Thom isomorphisms, and recover Al Amrani's extension to complex
K-theory. Our methods generalise to arbitrary complex oriented cohomology
algebras E^*(P(\chi)) and their dual homology coalgebras E_*(P(\chi)), as we
demonstrate for complex cobordism theory (the universal example). In
particular, we describe a fundamental class in \Omega^U_{2n}(P(\chi)), which
may be interpreted as a resolution of singularities.Comment: 26 page
Toric Genera
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 \Phi, which was introduced independently
by Krichever and Loeffler in 1974, albeit from radically different viewpoints.
In fact \Phi 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 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.Comment: 35 pages, LaTeX. In v2 references made to the index theoretical
approach to genera; rigidity and multiplicativity results improved;
acknowledgements adde
The equivariant -theory and cobordism rings of divisive weighted projective spaces
We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal
equivariant complex -theory ring of a divisive weighted projective space
(which is singular for nontrivial weights) is isomorphic to the ring of
integral piecewise Laurent polynomials on the associated fan. Analogues of this
description hold for other complex-oriented equivariant cohomology theories, as
we confirm in the case of homotopical complex cobordism, which is the universal
example. We also prove that the Borel versions of the equivariant -theory
and complex cobordism rings of more general singular toric varieties, namely
those whose integral cohomology is concentrated in even dimensions, are
isomorphic to rings of appropriate piecewise formal power series. Finally, we
confirm the corresponding descriptions for any smooth, compact, projective
toric variety, and rewrite them in a face ring context. In many cases our
results agree with those of Vezzosi and Vistoli for algebraic -theory,
Anderson and Payne for operational -theory, Krishna and Uma for algebraic
cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we
summarize the details of these coincidences.Comment: Accepted for publication in Tohoku Math.
Graduates of Character - Values and Character: Higher Education and Graduate Employment
Graduates of Character is the product of an empirical enquiry into the values, virtues, dispositions and attitudes of a sample of students and employees who volunteered to be involved. The research team sought host sites which would offer a diverse set of interviewees in gender, ethnicity, religion and aspiration.
In this study we discuss what character is taken to mean by students and employees in their years of higher education and employment. We examine what their values are, what they gain from the university, what they believe employers look for when recruiting, what they hope to give to an employer, and what they expect from their employer. We then explore who or what influenced their values and moral development. We also examined the role of the personal tutor or mentor, and the persons or services to which they might go for personal and/or professional support
Cost of delivering the early education entitlement : Research report
© Nigel Lloyd NLH Partnership Ltd
Flag manifolds and the Landweber-Novikov algebra
We investigate geometrical interpretations of various structure maps
associated with the Landweber-Novikov algebra S^* and its integral dual S_*. In
particular, we study the coproduct and antipode in S_*, together with the left
and right actions of S^* on S_* which underly the construction of the quantum
(or Drinfeld) double D(S^*). We set our realizations in the context of double
complex cobordism, utilizing certain manifolds of bounded flags which
generalize complex projective space and may be canonically expressed as toric
varieties. We discuss their cell structure by analogy with the classical
Schubert decomposition, and detail the implications for Poincare duality with
respect to double cobordism theory; these lead directly to our main results for
the Landweber-Novikov algebra.Comment: 23 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol2/paper5.abs.htm
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