708 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
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
The First-Order Genus of a Knot
We introduce a geometric invariant of knots in the three-sphere, called the
first-order genus, that is derived from certain 2-complexes called gropes, and
we show it is computable for many examples. While computing this invariant, we
draw some interesting conclusions about the structure of a general Seifert
surface for some knots.Comment: 14 pages, 17 figure
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