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We show that any preadditive infinity category with duality gives rise to a direct sum hermitian K-theory spectrum. This assignment is lax symmetric monoidal, thereby producing E-infinity ring spectra from preadditive symmetric monoidal infinity categories with duality. To have examples of preadditive symmetric monoidal infinity categories with duality we show that any preadditive symmetric monoidal infinity category, in which every object admits a dual, carries a canonical duality. Moreover we classify and twist the dualities in various ways and apply our definitions for example to finitely generated projective modules over E-infinity ring spectra., 26 pages

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology., Comment: Version closer to the published paper

We will give a definition of quadratic forms on bimodules and prove the sandwich classification theorem for subgroups of the general linear group $\mathrm{GL}(P)$ normalized by the elementary unitary group $\mathrm{EU}(P)$ if $P$ is a nondegenerate bimodule with large enough hyperbolic part., Comment: The replacement has been made, since I needed to include the gratitude to the sponsor of my institute. I also added a couple of links in the bibliography and fixed some misprints

Let $R$ be a commutative ring. For any projective $R$-module $P_0$ of constant rank $2$ with a trivialization of its determinant, we define a generalized Vaserstein symbol on the orbit space of the set of epimorphisms $P_0 \oplus R \rightarrow R$ under the action of the group of elementary automorphisms of $P_0 \oplus R$, which maps into the elementary symplectic Witt group. We give criteria for the surjectivity and injectivity of the generalized Vaserstein symbol and deduce that it is an isomorphism if $R$ is a regular Noetherian ring of dimension $2$ or a regular affine algebra of dimension $3$ over a perfect field $k$ with $c.d.(k) \leq 1$ and $6 \in k^{\times}$., Comment: 37 pages; final version; eliminated the assumption that 2 is invertible and updated the numbering to match the published version

We define a [math] -equivariant real algebraic [math] -theory spectrum [math] , for every [math] -equivariant spectrum [math] equipped with a compatible multiplicative structure. This construction extends the real algebraic [math] -theory of Hesselholt and Madsen for discrete rings, and the Hermitian [math] -theory of Burghelea and Fiedorowicz for simplicial rings. It supports a trace map of [math] -spectra [math] to the real topological Hochschild homology spectrum, which extends the [math] -theoretic trace of Bökstedt, Hsiang and Madsen. ¶ We show that the trace provides a splitting of the real [math] -theory of the spherical group-ring. We use the splitting induced on the geometric fixed points of [math] , which we regard as an [math] -theory of [math] -equivariant ring spectra, to give a purely homotopy theoretic reformulation of the Novikov conjecture on the homotopy invariance of the higher signatures, in terms of the module structure of the rational [math] -theory of the “Burnside group-ring”.

We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the $\mathcal{R}_{\bullet}$-construction from Grothendieck-Witt theory. We show that a relative $2$-Segal space defines a categorical representation of the Hall algebra associated to the base $2$-Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative $2$-Segal spaces., Comment: 45 pages. Final section split into two sections, with added details

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $V$, This invariant lies between $W(V)$ and the group $KO(V_{\mathbb R})$ of $\mathbb R$-linear topological vector bundles on $V_{\mathbb R}$, the set of real points of $V$. We show that the comparison maps $W(V)\to WR(V_{\mathbb C})$ and $WR(V_{\mathbb C})\to KO(V_{\mathbb R})$ that we define are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for the exponent of the kernel and cokernel of these maps, depending upon the dimension of $V.$ These results improve theorems of Knebusch, Brumfiel and Mah\'e. Along the way, we prove a comparison theorem between algebraic and topological Hermitian $K$-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.

Using an idea due to R.Thomason, we define a "homology theory" on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find P. Balmer's higher Witt groups. For more general rings, this homology isomorphic to the KT-theory of J. Hornbostel, inspired by the work of B. Williams. For real or complex C*-algebras, we recover - up to 2 torsion - topological K-theory., 6 pages ; see also http://www.math.jussieu.fr/~karoubi/

We prove that the stably free modules over a smooth affine threefold over an algebraically closed field of characteristic different from 2 are free., Comment: 11 pages

We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then \Gamma(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next consider the 3+1 dimensional cobordism 2-category of 3-spheres, 4-manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2-morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic K-theory of the integers. We also comment on versions of these results for simply connected spin 4-manifolds. Finally, we observe that a related 4-manifold operad detects infinite loop spaces., Comment: 22 pages, 3 figures; v2 - strengthened results on Dehn twists, added acknowledgements, corrected comments on relation to Hermitian K-theory; v3 - minor corrections and added section 8 on operads, final version to appear in Crelle's Journal; v4 - added reference

Let Λ be a ring endowed with an involution a 7→ a. We say that two units a and b of Λ fixed under the involution are congruent if there exists an element u ∈ Λ× such that a = ubũ. We denote by H(Λ) the set of congruence classes. In this paper we consider the case where Λ is an order with involution in a semisimple algebraA over a local field and study the question whether the natural map H(Λ)→ H(A) induced by inclusion is injective. We give sufficient conditions on the order Λ for this map to be injective and give applications to hermitian forms over group rings. Introduction and motivation Let R be a ring endowed with an involution : R → R (that is, an antiautomorphism of order 2). For a left R-module M we denote by M∗ the dual module HomR(M,R) with the left R-module structure given by (aφ)(m) = φ(m)a, for all a ∈ R, φ ∈ HomR(M,R), m ∈ M . Let ∈ R be a fixed central element satisfying = 1, for example = ±1. A (unimodular) -hermitian form over R is a pair (M, h) consisting of a reflexive R-module M and an isomorphism of R-modules h : M → M∗ satisfying h∗ = h. The notion of isometry of -hermitian forms is defined in the obvious way. It is a natural question to ask for a classification of -hermitian forms over R. An obvious necessary condition for two forms (M1, h1) and (M1 , h2) to be isometric is that their underlying R-modules M1 and M2 be isomorphic. This leads us to fix an R-module M and consider the set of all -hermitian forms on M . Assuming that this set is not empty, we fix once and for all an -hermitian form h0 : M → M∗ and we equip the endomorphism ring Λ = EndR(M) with the involution given by f = h−1 0 f h0. (1) A straightforward calculation shows that all the -hermitian forms on M are of the form h = h0a, with a ∈ Λ× satisfying a = a, and that two such forms h = h0a and g = h0b are isometric if and only if there exists u ∈ Λ× such that uaũ = b. Note that this is a particular case of the so-called transfer to the endomorphism ring in hermitian categories (see [15, Chapter 7, Section 4] or [13]). The above construction motivates the introduction of the following equivalence relation for any ring Λ equipped with an involution . a ∼ b ⇐⇒ there exists u ∈ Λ× such that uaũ = b The second–named author was supported by Louisiana Education Quality Support Fund grant No LEQSF(RF1995-97)-RD-A-40.