20,071 research outputs found
Zeros of closed 1-forms, homoclinic orbits, and Lusternik - Schnirelman theory
In this paper we study topological lower bounds on the number of zeros of
closed 1-forms without Morse type assumptions. We prove that one may always
find a representing closed 1-form having at most one zero. We introduce and
study a generalization of the notion of Lusternik - Schnirelman
category, depending on a topological space and a cohomology class . We prove that any closed 1-form has at least zeros
assuming that it admits a gradient-like vector field with no homoclinic cycles.
We show that the number can be estimated from below in terms of
the cup-products and higher Massey products. This paper corrects some
statements made in my previous papers on this subject.Comment: 34 pages. A refernce adde
Combinatorial invariants computing the Ray-Singer analytic torsion
It is shown that for any piecewise-linear closed orientable manifold of odd
dimension there exists an invariantly defined metric on the determinant line of
cohomology with coefficients in an arbitrary flat bundle E over the manifold (E
is not required to be unimodular). The construction of this metric (called
Poincare - Reidemeister metric) is purely combinatorial; it combines the
standard Reidemeister type construction with Poincare duality. The main result
of the paper states that the Poincare-Reidemeister metric computes
combinatorially the Ray-Singer metric. It is shown also that the Ray-Singer
metrics on some relative determinant lines can be computed combinatorially
(including the even-dimensional case) in terms of metrics determined by
correspondences.Comment: Amstex, 19 pages, to appear in "Differential Geometry and
Applications
Absolute torsion and eta-invariant
In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new
concept of combinatorial torsion which we called absolute torsion. Compared
with the classical Reidemeister torsion it has the advantage of having a
well-defined sign. Also, the absolute torsion is defined for arbitrary
orientable flat vector bundles, and not only for unimodular ones, as is
classical Reidemeister torsion.
In this paper I show that the sign behavior of the absolute torsion, under a
continuous deformation of the flat bundle, is determined by the eta-invariant
and the Pontrjagin classes.Comment: 10 page
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