The Whitehead Conjecture, the Tower of S^1 Conjecture, and Hecke algebras of type A

In the early 1980's the author proved G.W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990's, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd dimensional spheres. In this paper we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.Comment: 27 pages. As accepted for publication by the Journal of Topology. New: section 2 has been expanded, section 8 has been improved, and a dedication has been adde

Product and other fine structure in polynomial resolutions of mapping spaces

Let Map_T(K,X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum \Sigma^\infty Map_T(K,X). Applying a generalized homology theory h_* to this tower yields a spectral sequence, and this will converge strongly to h_*(Map_T(K,X)) under suitable conditions, e.g. if h_* is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on h_*(Map_T(K,X)). Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on Map_T(K,X) induces a `diagonal' on the associated tower. After applying any cohomology theory with products h^*, the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E_\infty -term corresponds to the cup product in h^*(Map_T(K,X)) in the usual way, and the product on the E_1-term is described in terms of group theoretic transfers. We use explicit equivariant S-duality maps to show that, when K is the sphere S^n, our constructions at the fiber level have descriptions in terms of the Boardman-Vogt little n-cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to \Sigma ^\infty \Omega ^\infty X.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-28.abs.htm

Understanding the truth about subjectivity

Results of two experiments show children’s understanding of diversity in personal preference is incomplete. Despite acknowledging diversity, in Experiment 1(N=108), 6- and 8-year-old children were less likely than adults to see preference as a legitimate basis for personal tastes and more likely to say a single truth could be found about a matter of taste. In Experiment 2 (N=96), 7- and 9-year-olds were less likely than 11- and 13-yearolds to say a dispute about a matter of preference might not be resolved. These data suggest that acceptance of the possibility of diversity does not indicate an adult-like understanding of subjectivity. An understanding of the relative emphasis placed on objective and subjective factors in different contexts continues to develop into adolescence

Librarians use of Web 2.0 in UK Medical Schools: Outcomes of a national survey

Using the results of an Email survey, this paper reviews the use of Web 2.0 technologies by librarians working in UK Medical Schools. Web 2.0 has been hailed as an innovation for facilitation of two way communication on the net, and it is, therefore, timely to measure how effectively librarians are capturing this opportunity for increased student engagement. The social nature of Web 2.0 can be particularly appropriate for undergraduate medical students who fit their studies around the unsocial hours and geographical isolation of clinical placements. This paper will investigate library use of blogs, Facebook, and Twitter. Consideration will also be given as to whether they facilitate a more collabroative library service or if they leave undergraduate medical students swamped with yet more information to manage