112 research outputs found
Finite type invariants of 3-manifolds
A theory of finite type invariants for arbitrary compact oriented 3-manifolds
is proposed, and illustrated through many examples arising from both classical
and quantum topology. The theory is seen to be highly non-trivial even for
manifolds with large first betti number, encompassing much of the complexity of
Ohtsuki's theory for homology spheres. (For example, it is seen that the
quantum SO(3) invariants, though not of finite type, are determined by finite
type invariants.) The algebraic structure of the set of all finite type
invariants is investigated, along with a combinatorial model for the theory in
terms of trivalent "Feynman diagrams".Comment: Final version for publication, with figures. The most significant
changes from the original posted version are in the exposition of section 3
(on the Conway polynomial) and section 4 (on quantum invariants
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
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