Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders

We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo H. To any unitary representation, r of pi_1(S)/H we associate a higher-order rho_r-invariant and a signature 2-cocycle sigma_r. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each rho_r is a quasimorphism and each sigma_r is a bounded 2-cocycle on J(H). In one of the simplest non-trivial cases, by varying r, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the rho_r restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on S.Comment: 38 pages. This is final version for publication in IMRN, deleted some material and many references (sorry-at referee's insistence

Marine Protected Areas: Economic and Social Implications

This paper is a guide for citizens, scientists, resource managers, and policy makers, who are interested in understanding the economic and social value of marine protected areas (MPAs). We discuss the potential benefits and costs associated with MPAs as a means of illustrating the economic and social tradeoffs inherent in implementation decisions. In general, the effectiveness of a protected area depends on a complex set of interactions between biological, economic, and institutional factors. While MPAs might provide protection for critical habitats and cultural heritage sites and, in some cases, conserve biodiversity, as a tool to enhance fishery management their impact is less certain. The uncertainty stems from the fact that MPAs only treat the symptoms and not the fundamental causes of overfishing and waste in fisheries.Marine Protected Areas (MPAs), marine reserves, fisheries

Longinus, Sappho’s Ode, and the Question of Sublimity

De prime abord, cet article peut sembler porter sur les attitudes à l’endroit de l’autorité des anciens, mais il porte en fait sur l’ironie du ton et sur la difficulté qu’on peut avoir à la déceler, même quand le locuteur ou l’écrivain est un de nos proches. Dans Don Juan, chant premier, strophe 42, par exemple, Byron écrit : « je ne crois pas que l’ode de Sapho soit d’un bon exemple, quoique Longin prétende qu’il n’est point d’hymne où le sublime prenne un essor plus élevé . . .1 », suite à quoi il cite des passages du pseudo-Longin auquel il fait référence. Dans les marges de l’épreuve, J. C. Hobhouse le corrige — ou du moins il tente de le faire — en proposant une autre interprétation de ce qu’entend Longin, différente de celle que communique la strophe de Byron. Dans les marges de la même épreuve, Byron réagit « robustement » comme il avait parfois l’habitude de le faire et refuse d’apporter les modifications proposées.Dans cet article, j’examinerai les deux attitudes qu’illustre le micro-argument entre Hobhouse et Byron à l’égard de l’autorité classique, et je verrai ce qu’on peut en déduire sur la difficulté qu’avaient les tout premiers lecteurs de Don Juan à cerner le ton de Byron et son attitude à l’égard de l’autorité et au précédent. Ce faisant, j’espère nous donner une idée de ce qu’aurait pu signifier le terme « sublime » pour (1) Byron avant Don Juan, (2) Byron à l’ère de Don Juan, et (3) un lecteur conservateur comme Hobhouse (qui représente le lecteur averti moyen vers 1819). J’examinerai également ce qu’aurait pu être la signification de l’Ode de Sapho pour chacun de ces deux hommes, et je me demanderai si la nature même de l’ode a une incidence sur notre perception de ce qu’entend Byron en parlant du « Sublime »

Knot concordance and homology cobordism

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.Comment: 15 pages, 8 figure

Tree homology and a conjecture of Levine

In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasi-Lie algebra. Both T and D' have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory, and the homology of the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying the structure of links up to grope and Whitney tower concordance, as explained in other papers of this series. We also confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders

Higher order intersection numbers of 2-spheres in 4-manifolds

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions. Our second order obstruction tau(f) is defined if mu(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of pi_1(X) modulo S_3-symmetry (rather then just one copy modulo S_3-symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sufficient conditions for moving three maps f_1,f_2,f_3:S^2 --> X^4 to a position in which they have disjoint images. Again the obstruction lambda(f_1,f_2,f_3) generalizes Wall's intersection number lambda(f_1,f_2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant mu(1,2,3), generalizing the Matsumoto triple to the non simply-connected setting.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-1.abs.htm