477 research outputs found
Complex valued Ray-Singer torsion
In the spirit of Ray and Singer we define a complex valued analytic torsion
using non-selfadjoint Laplacians. We establish an anomaly formula which permits
to turn this into a topological invariant. Conjecturally this analytically
defined invariant computes the complex valued Reidemeister torsion, including
its phase. We establish this conjecture in some non-trivial situations.Comment: Fixed two sign mistakes and added a few more details here and ther
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
Refinement of Novikov - Betti numbers and of Novikov homology provided by an angle valued map
To a pair (X,f), X compact ANR and f a continuous angle valued map defined on X, a fixed field and a nonnegative integer one assigns a finite configuration of complex numbers with multiplicities located in the punctured complex plane and a finite configuration of free modules over the ring of Laurent polynomials (with coefficients in the fixed field) indexed by the same complex numbers. This is done in analogy with the configuration of eigenvalues and of generalized eigenspaces of an invertible linear operator in a finite dimensional complex vector space. The configuration of complex numbers refines the Novikov - Betti number and the configuration of free modules refines the Novikov homology associated with the cohomology class defined by f, in the same way the collection of eigenvalues and of generalized eigen-spaces refine the dimension of the vector space and the vector space on which the operator acts. In the case the field is the field of complex numbers the configuration of free modules induces by "von-Neumann completion" a configuration of mutually orthogonal closed Hilbert submodules of the L 2--homology of the infinite cyclic cover of X determined by the map f, which is an Hilbert module over the von-Neumann algebra of complex L-infinity functions on the unit circle in the complex plane
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 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
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