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

Abstract

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.

38 pages. This is final version for publication in IMRN, deleted some material and many references (sorry-at referee's insistence)

References in corpus (8)

• Primary decomposition and the fractal nature of knot concordance
• 2-torsion in the n-solvable filtration of the knot concordance group
• Quasi-homomorphisms and stable lengths in mapping class groups
• Infinite generation of the kernels of the Magnus and Burau representations
• The mapping class group and the Meyer function for plane curves
• The L^2 signature of torus knots
• Abelian covers of surfaces and the homology of the level L mapping class group
• On Rho invariants of fiber bundles

Cited by in corpus (7)

• On stable commutator length in hyperelliptic mapping class groups
• $\hat{G}$-invariant quasimorphisms and symplectic geometry of surfaces
• Invariants and structures of the homology cobordism group of homology cylinders
• A new filtration of the Magnus kernel of the Torelli group
• Coloring Link Diagrams And Conway-type Polynomial Of Braids
• The space of non-extendable quasimorphisms
• Homology cobordism group of homology cylinders and invariants related to lower central series