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Dualization in algebraic K-theory and the invariant e¹ of quadratic forms over schemes

Marek Szyjewski (2011)

Fundamenta Mathematicae

In the classical Witt theory over a field F, the study of quadratic forms begins with two simple invariants: the dimension of a form modulo 2, called the dimension index and denoted e⁰: W(F) → ℤ/2, and the discriminant e¹ with values in k₁(F) = F*/F*², which behaves well on the fundamental ideal I(F)= ker(e⁰). Here a more sophisticated situation is considered, of quadratic forms over a scheme and, more generally, over an exact category with duality. Our purposes are: ...

Elliptic operators and higher signatures

Eric Leichtnam, Paolo Piazza (2004)

Annales de l’institut Fourier

Building on the theory of elliptic operators, we give a unified treatment of the following topics: - the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds, - the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds, - the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.

Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup

Jon F. Carlson, Nadia Mazza, Jacques Thévenaz (2013)

Journal of the European Mathematical Society

Let G be a finite group with a Sylow 2-subgroup P which is either quaternion or semi-dihedral. Let k be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial k G -modules, whose restrictions to P are isomorphic to the direct sum of the known exotic endotrivial k P -modules and some projective modules. This provides a description of the group T ( G ) of endotrivial k G -modules.

Enlacements d’intervalles et torsion de Whitehead

Jean-Yves Le Dimet (2001)

Bulletin de la Société Mathématique de France

Soit E un enlacement de n intervalles dans D 2 × I d’extérieur X et soit X 0 = X ∩ D 2 × 0 . On utilise la propriété de la paire ( X , X 0 ) d’être Λ -acyclique pour certaines représentation ρ de l’anneau du groupe fondamental π de X dans un anneau Λ pour construire des invariants de torsion à valeurs dans le groupe K 1 ( Λ ) / ρ ( ± π ) . Un cas particulier est le polynôme d’Alexander en n variables quand Λ est l’anneau des fractions rationnelles P / Q avec Q ( 1 , 1 , ⋯ , 1 ) = 1 et ρ est simplement l’abélianisation.

Equations for Mahler measure and isogenies

Matilde N. Lalín (2013)

Journal de Théorie des Nombres de Bordeaux

We study some functional equations between Mahler measures of genus-one curves in terms of isogenies between the curves. These equations have the potential to establish relationships between Mahler measure and especial values of L -functions. These notes are based on a talk that the author gave at the “Cuartas Jornadas de Teoría de Números”, Bilbao, 2011.

Equivariant K-theory of flag varieties revisited and related results

V. Uma (2013)

Colloquium Mathematicae

We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring K T ( G / B ) of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in K T ( G / B ) to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of K ( X ) ℚ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of...

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