281 research outputs found
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
The Antiretroviral Pipeline
The year 2016 marks 20 years since combination-based antiretroviral therapy (ART) first demonstrated durable, effective and sustained HIV control. An unprecedented period of drug discovery followed, and advances in viral load and resistance technology made HIV, in high-income countries, one of the most individualized infections to manage.This chapter on Antiretroviral Treatment in this year's Pipeline Report provides a sweeping overview of developments in the past twenty years to put those of the past year into context
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
Filtering smooth concordance classes of topologically slice knots
We propose and analyze a structure with which to organize the difference
between a knot in the 3-sphere bounding a topologically embedded 2-disk in the
4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of
the topological knot concordance group, due to Cochran-Orr-Teichner, may be
complete in the sense that any knot in the intersection of its terms may well
be topologically slice. However, the natural extension of this filtration to
what is called the n-solvable filtration of the smooth knot concordance group,
is unsatisfactory because any topologically slice knot lies in every term of
the filtration. To ameliorate this we investigate a new filtration, {B_n}, that
is simultaneously a refinement of the n-solvable filtration and a
generalization of notions of positivity studied by Gompf and Cochran. We show
that each B_n/B_{n+1} has infinite rank. But our primary interest is in the
induced filtration, {T_n}, on the subgroup, T, of knots that are topologically
slice. We prove that T/T_0 is large, detected by gauge-theoretic invariants and
the tau, s, and epsilon-invariants; while the non-triviliality of T_0/T_1 can
be detected by certain d-invariants. All of these concordance obstructions
vanish for knots in T_1. Nonetheless, going beyond this, our main result is
that T_1/T_2 has positive rank. Moreover under a "weak homotopy-ribbon"
condition, we show that each T_n/T_{n+1} has positive rank. These results
suggest that, even among topologically slice knots, the fundamental group is
responsible for a wide range of complexity.Comment: 41 pages, slightly revised introduction, minor corrections and
up-dated references, this is the final version to appear in Geometry and
Topolog
Genomic DNA k-mer spectra: models and modalities
Tetrapods, unlike other organisms, have multimodal spectra of k-mers in their genome
- …