Geometric cycles and manifold bundles
Much recent work has gone toward computing the ring H*( BDiff(M) ) of characteristic classes of manifold bundles. This is a difficult problem, but in many cases we have a good understanding of the cohomology in the "stable range". In this paper we produce classes outside the stable range coming from the unstable part of the cohomology of arithmetic groups.
On groups whose Bowditch boundary is the 2-sphere
(with Genevieve Walsh)
Bestvina-Mess showed that the duality properties of a group are encoded in any boundary that gives a Z-set compactification; in particular, a hyperbolic group with Gromov boundary S^n is a PD(n+1) group. For relatively hyperbolic pairs (G,P), the Bowditch boundary does not give a Z-set compactification. Nevertheless we show that if the Bowditch boundary of (G,P) is S^2, then (G,P) is a PD(3) pair.
Characteristic classes of fiberwise branched surface bundles via arithmetic groups
Let S be a closed surface of genus g>1. The cohomology of its mapping class group Mod(S) and its finite-index subgroups plays an important role in many areas of mathematics. One source of cohomology classes is the symplectic representation
R: Mod(S) --> Sp(2g,Z). A well-known computation shows that the image of
R*: H*( Sp(2g,Z) ;Q) --> H*( Mod(S) ;Q) is the algebra generated by the odd Miller-Morita-Mumford (MMM) classes in the stable range. In this paper we extend this example. Let G < Diff(S) be a finite group, and consider the subgroup Mod^G(S) of Mod(S) of mapping classes that can be realized by diffeomorphisms that commute with G. The image of Mod^G(S) under R lands in the centralizer Sp^G(2g,Z). The main result of this paper is a computation of R*: H^2( Sp^G(2g,Z) ;Q) --> H^2( Mod^G(S) ;Q)
when G is a finite cyclic group. We apply this computation to compute Toledo invariants of surface group representations to SU(p,q) arising from the Atiyah-Kodaira construction, and we show that classes in the image of R* gives equivariant cobordism invariants for surface bundles with fiberwise G action, following Church-Farb-Thibault.
On the non-realizability of braid groups by diffeomorphisms
(with Nick Salter)
Bull. Lond. Math. Soc. 2016
For every compact surface we show that when n is sufficiently large there is no lift of the surface braid group Br(n,S) to Diff(S,n), the group of diffeomorphisms preserving n marked points and restricting to the identity on the boundary. This generalizes work of Bestvina-Church-Souto, but our main tool (the Thurston stability theorem) is different. Our methods are applied to give a new proof of Morita's non-lifting theorem in the best possible range. These techniques extend to the more general setting of spaces of codimension-2 embeddings, and we obtain corresponding results for spherical motion groups, including the string motion group.
Hyperbolic groups with boundary an n-dimensional Sierpinski space
(with Jean Lafont)
J. Topol. Anal. to appear
The fundamental group of a closed aspherical manifold M is an example of a Poincare duality group. Whether or not all finitely presented Poincare duality groups arise in this fashion is an open problem that goes back to Wall. This paper addresses a relative version of this problem for a special class of groups. Fixing n>6, we show that if G is a torsion-free hyperbolic group whose visual boundary is an (n-2)-dimensional Sierpinski space, then G = pi_1(W) for some aspherical n-manifold W with boundary. The proof involves the total surgery obstruction and is based on work of Bartels-Lueck-Weinberger. Concerning the converse, we construct examples of aspherical manifolds with boundary whose fundamental group is hyperbolic but with visual boundary not homeomorphic to a Sierpinski space.
Pontryagin classes of locally symmetirc spaces
Algebr. Geom. Topol. 2015
In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class? This computation was motivated by a Nielsen realization problem. Specifically, in my paper "Cohomological obstructions to Nielsen realization", I show that Pontryagin classes are obstructions to realizing the point-pushing subgroup for nonpositively curved manifolds (and in particular locally symmetric manifolds of noncompact type).
Obstructions to Nielsen realization
J. Topol. 2015
For a manifold M and a subgroup G < Mod(M) of the mapping class group, the question of whether G can be lifted to the diffeomorphism group Diff(M) is an example of a Nielsen realization problem. This question is related to an existence question for flat connections on fiber bundles with with monodromy G. In this paper we consider the case M is a locally symmetric manifold with a basepoint and G is the point-pushing subgroup. The simplest instance -- when M is a closed orientable surface of genus at least 2 -- was worked out by Bestvina-Church-Souto, who showed a lift does not exist. We give new cohomological techniques to generalize their result to higher dimensional locally symmetric manifolds. The main tools include Chern-Weil theory, Milnor-Wood inequalities, and Margulis superrigidity.