A refinement of Betti numbers and homology in the presence of a continuous function II (the case of an angle valued map)

For a continuous angle-valued map defined on a compact ANR, a fixed field and any degree one proposes a refinement of the Novikov-Betti number and of the Novikov homology of the pair consisting of the ANR and the degree one integral cohomology class represented by the map. For each degree the first refinement is a configuration of points with multiplicity located in the punctured complex plane whose total cardinality is the Novikov-Betti number of the pair. The second refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and which has the same support as the first configuration. When the field is a the field of complex numbers the second configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of the ANR defined by the angle-valued map. One discusses the properties of these configurations namely, robustness with respect to continuous perturbation of the angle-valued map and the Poincar\'e Duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov-Betti numbers replacing standard homology and standard Betti numbers.Comment: 38 page

A refinement of Betti numbers in the presence of a continuous function. ( I )

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of vector spaces indexed by points in complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product these vector spaces are canonically realized as mutually orthogonal subspaces of the homology. The assignments above are in analogy with the collections of eigenvalues and generalized eigenspaces of a linear map in a finite dimensional complex vector space. A number of remarkable properties of the above configurations are discussed.Comment: 24 page

Lectures on Witten Helffer Sj\"ostrand Theory

Witten- Helffer-Sj\"ostrand theory is a considerable addition to the De Rham- Hodge theory for Riemannian manifolds and can serve as a general tool to prove results about comparison of numerical invariants associated to compact manifolds analytically, i.e. by using a Riemannian metric, or combinatorially, i.e by using a triangulation. In this presentation a triangulation, or a partition of a smooth manifold in cells, will be viewed in a more analytic spirit, being provided by the stable manifolds of the gradient of a nice Morse function. WHS theory was recently used both for providing new proofs for known but difficult results in topology, as well as new results and a positive solution for an important conjecture about $L_2-$torsion, cf [BFKM]. This presentation is a short version of a one quarter course I have given during the spring of 1997 at OSU.Comment: 17 pages, AMStex, minor grammar correction