Mathematics

In this paper, we consider the Hermitian K -theory of schemes with involution, for which we construct a transfer morphism and prove a version of the dévissage theorem. This theorem is then used to compute the Hermitian K -theory of P 1 with involution given by [X:Y]↦[Y:X] . We also prove the C 2 -equivariant $\A^1$-invariance of Hermitian K -theory, which confirms the representability of Hermitian K -theory in the C 2 -equivariant motivic homotopy category of Heller, Krishna and \{O}stvær \cite{HKO14}.

In this paper we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end. We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary. This paves the way for classification of these metrics. Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.

A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ZG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ZG has SFC provided at most one copy of the quaternions H occurs in the Wedderburn decomposition of the real group ring RG . This generalises the Eichler condition in the case of integral group rings.

We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf algebras.

We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's S ∙ -construction and therefore it defines a model for Waldhausen K -theory. As an example, we discuss this model for A -theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's A(∗) . We also review the canonical map from the cobordism category of manifolds to A -theory from this viewpoint.

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the Fredholm module.

We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr-Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.

Our (weak) conjecture claims that a finite dimensional Lie algebra {\bf g} over the field of complex numbers is semi-simple iff the Leibniz homology vanishes in positive dimensions HL_i({\bf g})=0 , i>0 . We will indicate a mistake in the recent proof of this conjecture due to Burde and Wagemann.

In a 2006 article Schlichting conjectured that the negative {\it K--}theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative {\it K--}theory of any category with a bounded {\it t--}structure must vanish. Both conjectures will be shown to be false.

In this short note, we given a new proof of Mitchell's theorem that L T(n) K(Z)≅0 for n≥2 . Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology for chromatically-localized algebraic K-theory.