Approximating L^2-signatures by their compact analogues

:Let G be a group together with an descending nested sequence of normal subgroups G=G_0, G_1, G_2 G_3, ... of finite index [G:G_k] such the intersection of the G_k-s is the trivial group. Let (X,Y) be a compact 4n-dimensional Poincare' pair and p: (\bar{X},\bar{Y}) \to (X,Y) be a G-covering, i.e. normal covering with G as deck transformation group. We get associated $G/_k$-coverings (X_k,Y_k) \to (X,Y). We prove that sign^{(2)}(\bar{X},\bar{Y}) = lim_{k\to\infty} \frac{sign(X_k,Y_k)}{[G : G_k]}, where sign or sign^{(2)} is the signature or L^2-signature, respectively, and the convergence of the right side for any such sequence (G_k)_k is part of the statement

Maslov index, Lagrangians, Mapping Class Groups and TQFT

Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.Comment: 31 pages, 11 Figures. to appear in Forum Mathematicu

Factor maps between tiling dynamical systems

We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such systems which cannot be achieved by working within a finite window. By considering 1-dimensional tiling systems, which are the same as flows under functions on subshifts with finite alphabets of symbols, we construct a `simple' code which is not `local', a local code which is not simple, and a continuous code which is neither local nor simple.Comment: 8 page

Stratified fibre bundles

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem.Comment: LaTeX file. Revised version. Accepted for publication on "Forum Mathematicum

Manifolds associated with (Z2)n(Z_2)^n-colored regular graphs

In this article we describe a canonical way to expand a certain kind of $(\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\leq 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a $(\mathbb Z_2)^{n+1}$-colored regular graph admits an $n$-skeletal expansion, then it is realizable as the moment graph of an $(n+1)$-dimensional closed $(\mathbb Z_2)^{n+1}$-manifold.Comment: 20 pages with 9 figures, in AMS-LaTex, v4 added a new section on reconstructing a space with a $(Z_2)^n$-action for which its moment graph is a given colored grap

The Lerch Zeta Function II. Analytic Continuation

This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. In this paper we analytically continue it as a function of three complex variables. We that it is well defined as a multivalued function on the manifold M equal to C^3 with the hyperplanes corresponding to integer values of the two variables a and c removed. We show that it becomes single valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of their properties.Comment: 29 pages, 3 figures; v2 notation changes, homotopy action on lef