Homotopy Lie groups

Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson, represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector in his vision of a homotopy theoretic incarnation of Lie group theory. What was then technically impossible has now become feasible thanks to modern advances such as Miller's proof of the Sullivan conjecture and Lannes's division functors. Today, with Dwyer and Wilkerson's implementation of Rector's vision, the tantalizing classification theorem seems to be within grasp. Supported by motivating examples and clarifying exercises, this guide quickly leads, without ignoring the context or the proof strategy, from classical finite loop spaces to the important definitions and striking results of this new theory.Comment: 16 page

Homotopy equivalences between p-subgroup categories

Let p be a prime number and G a finite group of order divisible by p. Quillen showed that the Brown poset of nonidentity p-subgroups of G is homotopy equivalent to its subposet of nonidentity elementary abelian subgroups. We show here that a similar statement holds for the fusion category of nonidentity p-subgroups of G. Other categories of p-subgroups of G are also considered.Comment: 19 pages. Second versio

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $\chi_2$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere $S^3$ with face vector $f=(167,1579,2824,1412)$ and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

A person-time analysis of hospital activity among cancer survivors in England.

BACKGROUND: There are around 2 million cancer survivors in the UK. This study describes the inpatient and day case hospital activity among the population of cancer survivors in England. This is one measure of the burden of cancer on the individual and the health service. METHODS: The national cancer registry data set for England (1990-2006) is linked to the NHS Hospital Episode Statistics (HES) database. Cohorts of survivors were defined as those people recorded in the cancer registry data with a diagnosis of breast, colorectal, lung or prostate cancer before 2007. The person-time of prevalence in 2006 for each cohort of survivors was calculated according to the cancer type, sex, age and time since diagnosis. The corresponding HES episodes of care in 2006 were used to calculate the person-time of admitted hospital care for each cohort of survivors. The average proportion of time spent in hospital by survivors in each cohort was calculated as the summed person-time of hospital activity divided by the summed person-time of prevalence. The analysis was conducted separately for cancer-related episodes and non-cancer-related episodes. RESULTS: Lung cancer survivors had the highest intensity of cancer-related hospital activity. For all cancers, cancer-related hospital activity was highest in the first year following diagnosis. Breast and prostate cancer survivors had peaks of cancer-related hospital activity in the relatively young and relatively old age groups. The proportion of time spent in hospital for non-cancer-related care was much lower than that for cancer-related care and increased gradually with age but was generally constant regardless of time since diagnosis. CONCLUSION: The person-time approach used in this study is more revealing than a simple enumeration of cancer survivors and hospital admissions. Hospital activity among cancer survivors is highest soon after diagnosis. The effect of age on the amount of hospital activity is different for each type of cancer