Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)

We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this integrality by proving that the complex of partial bases of O^n is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^{vcd}(SL_n(O_K); Q), where O_K is the ring of integers in a number field and vcd is the virtual cohomological dimension of SL_n(O_K). The (non)vanishing depends on the (non)triviality of the class group of O_K. We also obtain a vanishing theorem for the cohomology H^{vcd}(SL_n(O_K); V) with twisted coefficients V.Comment: 36 pages; final version; to appear in Amer. J. Mat

A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)

In this paper we conjecture the stability and vanishing of a large piece of the unstable rational cohomology of SL_n Z, of mapping class groups, and of Aut(F_n).Comment: 18 pages. v2: final version, to appear in Algebraic Topology: Applications and New Directions, AMS Contemporary Mathematics Serie

Application of Investigational Device Exemptions regulations to endograft modification

For patients with complex aortic aneurysms that cannot be treated effectively with currently approved endografts, physicians have made fenestrations in marketed devices and constructed branched grafts by creatively implanting available endograft components. For the most part, these procedures are being done outside of clinical studies by individual physicians. Although these novel approaches may be useful in the treatment of individual patients, the current ad hoc use of physician-created fenestrated and branched devices may not result in the unbiased capture and reporting of data regarding short- and longer-term outcomes. As a result, unsubstantiated conclusions regarding the safety and effectiveness of these procedures may be drawn. Well-designed and executed clinical studies are necessary to adequately assess the benefits and risks of these techniques. Because these interventions involve the use of significant risk devices, these studies need to be conducted under United States Food and Drug Administration-approved Investigational Device Exemptions (IDE) applications. Although this regulatory process adds complexity to the application of these creative techniques, the IDE regulations assure that patient protection measures are followed and data are captured to assess safety and effectiveness. This approach creates opportunities to advance the development of innovative, beneficial devices and procedures to treat complex aortic aneurysms

The Casson invariant and the word metric on the Torelli group

We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group \T, of the element of \T associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp.Comment: 5 pages, minor corrections; to appear in C. R. Math. Acad. Sci. Pari

Small generating sets for the Torelli group

Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space called the handle graph on which the Torelli group acts cocompactly.Comment: 12 pages, 3 figures; serious revision; to appear in Geom. Topo

The rational cohomology of the mapping class group vanishes in its virtual cohomological dimension

Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^{4g-5}(Mod_g; Q) = 0.Comment: 4 pages. Final version, to appear in IMR