Let M be a closed COO -manifold of dimension n. Both R-torsion and analytic torsion are smooth invariants of acyclic orthogonal (or unitary) representations P of the fundamental group 1f l (M). The Reidemeister-Franz torsion (or R-torsion) r M(P) of P is defined in terms of the combinatorial structure of M given by its smooth triangulations. The analytic torsion T M(P) was introduced by Ray and...

We compare the higher analytic torsion T of Bismut and Lott of a fibre bundle p:M → B equipped with a flat vector bundle F → M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F , thereby extending previous joint work with Bismut. Under additional conditions on F , the torsion T is related to Igusa’s...

We give an introduction to the theory of determinantal ideals and rings, their Gröbner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the Knuth-Robinson-Schensted correspondence. The article contains a section treating the basic results about the passage to initial ideals and algebras. Let K be a field and X an m×n matrix of indeterminates o...

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments. The classical Franz-Reidemeister torsion τFR is an invariant of manifolds with acyclic unitarily flat vector bundles [62], [33]. In contrast to most other algebraic-topological invariants known at that time, it is invariant under homeomorp...

Mathai [M] has conjectured that the Cheeger-Gromov invariant ρ(2) = η(2) − η is a homotopy invariant of closed manifolds with torsion-free fundamental group. In this paper we prove this statement for closed manifolds M when the rational Borel conjecture is known for Γ = π1(M), i.e. the assembly map α : H∗(BΓ,Q)→ L∗(Γ)⊗Q is an isomorphism. Our discussion evokes the theory of intersection homolog...

We compare the higher analytic torsion T of Bismut and Lott of a fibre bundle p:M → B equipped with a flat vector bundle F → M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F , thereby extending previous joint work with Bismut. Under additional conditions on F , the torsion T is related to Igusa’s...

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U ⊂ M , based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field γ defining a parallelism structure on U ⊂ M , which represents in...

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U ⊂ M , based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field γ defining a parallelism structure on U ⊂ M , which represents in...