Juergen Eichhorn | Searcharxiv

arXiv2001

Relative Zeta Functions, Determinants, Torsion, Index Theorems and Invariants for Open Manifolds

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac operators, we prove for the pair $(D,D')$ relative index theorems, define relative $ζ$-- and $η$--functions, relative determinants and in the case of $D=Δ$ relative analytic torsion. To define relative $ζ$-- and $η$--functions, we assume additionally that the essential spectrum of $D^2$ has a gap above zero.

math.DG↗

arXiv1999

A Classification Approach for Open Manifolds

There are a least uncountably many diffeomorphism types for open manifolds. Hence the classification problem is extremely difficult. We proceed as follows: We define several uniform structures of proper metric spaces and consider their arc components. Any open complete manifold (M^n,g) defines such a component. Hence the classification amounts to two steps. First counting all components, and secondly counting all elements inside a component. This can be partially done by invariants. We define bordism groups and relative characteristic numbers.

math.DG↗