A Braided Simplicial Group

By studying the braid group action on Milnor's constructionof the 1-sphere, we show that the general higher homotopy groupof the 3-sphere is the fixed set of the pure braid group actionon certain combinatorially described groups. This establishesa relation between the braid groups and the homotopy groupsof the sphere. 2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05,55Q40, 55U10.     

p-Adic Automorphic Functions for the Unitary Group in Three Variables

Let G be a special unitary group in three variables over a local field and, a co-compact subgroup of G. We construct, -automorphic functions on the rigid analytic space of points in the projective plane stable under the action of tori in G.This research was partially supported by an ARC grant.2000 Mathematics Subject Classification: 11F55, 11F85, 20G25, 22E35     

Homotopy equivalences of p-completed classifying spaces of finite groups

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category ℒ _p ^c (G) with the following properties. Two p-completed classifying spaces BG _p ^∧ and BG’ _p ^∧ have the same homotopy type if and only if the associated categories ℒ _p ^c (G) and ℒ _p ^c (G’) are equivalent. Furthermore, the topological monoid Aut(BG _p ^∧) of self equivalences is determined by the self equivalences of the associated category ℒ _p ^c (G). Oblatum 5-VII-2001 & 28-VIII-2002?Published online: 8 November 2002 RID="*" ID="*"C. Broto is partially supported by DGICYT grant PB97–0203. RID="**" ID="**"R. Levi is partially supported by EPSRC grant GR/M7831. RID="***" ID="***"B. Oliver is partially supported by UMR 7539 of the CNRS.     

Quiver Grassmannians associated with string modules

We provide a technique to compute the Euler–Poincaré characteristic of a class of projective varieties called quiver Grassmannians. This technique applies to quiver Grassmannians associated with “orientable string modules”. As an application we explicitly compute the Euler–Poincaré characteristic of quiver Grassmannians associated with indecomposable pre-projective, pre-injective and regular homogeneous representations of an affine quiver of type [(A)\tilde]_p,1\tilde{A}_{p,1}. For p=1, this approach provides another proof of a result due to Caldero and Zelevinsky (in Mosc. Math. J. 6(3):411–429, 2006).     

Symmetric Brace Algebras

We develop a symmetric analog of brace algebras and discuss the relation of such algebras to L_∞-algebras. We give an alternate proof that the category of symmetric brace algebras is isomorphic to the category of pre-Lie algebras. As an application, symmetric braces are used to describe transfers of strongly homotopy structures. We then explain how these symmetric brace algebras may be used to examine the L_∞-algebras that result from a particular gauge theory for massless particles of high spin.Mathematics Subject Classifications (2000) 55S20 (primary), 70S15 (secondary).Tom Lada: The research of the first author was supported in part by NSF grant INT-0203119.Martin Markl: The research of the second author was supported by grant MŠMT ME 603 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Asymptotic coarsening of proper metric spaces and index problems

 We introduce an asymptotic coarse structure on proper metric spaces and study the associated C ^* -algebras and assembly maps. We establish an asymptotic Lipschitz homotopy invariance theorem for the K-theory of these C ^* -algebras and the K-homology of the metric space, and show that the assembly map is an isomorphism over an asymptotically scaleable space. Received: 12 July 2001 / Revised version: 29 October 2002 Published online: 3 March 2003 The first author is supported in part by the National Basic Research Project (973), NSF and the Educational Ministry of P. R. China. The second author is supported by the NSF grant 10201007, P. R. China, and a grant from Shanghai Science and Technology Commission, No. 01ZA14003. Mathematics Subject Classification (2000): Primary 46L80     

On fixed points of permutations

The number of fixed points of a random permutation of {1,2,…,n} has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial – almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of {1,2,…,n}, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results. This paper is dedicated to the life and work of our colleague Manfred Schocker. We thank Peter Cameron for his help. Diaconis was supported by NSF grant DMS-0505673. Fulman received funding from NSA grant H98230-05-1-0031 and NSF grant DMS-0503901. Guralnick was supported by NSF grant DMS-0653873. This research was facilitated by a Chaire d’Excellence grant to the University of Nice Sophia-Antipolis.     

On the homotopy groups of p-completed classifying spaces

Among the generalizations of Serre's theorem on the homotopy groups of a finite complex we isolate the one proposed by Dwyer and Wilkerson. Even though the spaces they consider must be 2-connected, we show that it can be used to both recover known results and obtain new theorems about p-completed classifying spaces. All three authors are partially supported by MEC grant MTM2004-06686. The third author is supported by the program Ramón y Cajal, MEC, Spain, and thanks the CIB (Centre Interfacultaire Bernoulli), EPFL, Lausanne for its hospitality.     

Free energy of the spherical mean field model

We compute at any temperature the free energy of the multi p-spin spherical model when only terms for p even are considered. Work partially supported by an NSF grant     

Orbits of Groups Generated by Transvections over F<Subscript>2</Subscript>

Let V be a finite dimensional vector space over the two element field. We compute orbits for the linear action of groups generated by transvections with respect to a certain class of bilinear forms on V. n particular, we compute orbits that are in bijection with connected components of real double Bruhat cells in semisimple groups, extending results of M. Gekhtman, B. Shapiro, M. Shapiro, A. Vainshtein and A. Zelevinsky.The author’s research was supported in part by Andrei Zelevinsky’s NSF grant #DMS-0200299.     

Cartan determinants and Shapovalov forms

We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of p-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity. Received: 20 November 2001 / Revised version: 7 February 2002 / Published online: 27 June 2002 Second author partially supported by the NSF (grant no. DMS-9900134).     

The group of homotopy equivalences of products of spheres and of Lie groups

We investigate the group of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups. We obtain results on the structure of provided the p-localization of X has the homotopy type of a p-local product of odd-dimensional spheres. In particular, we show that is a semidirect product of certain homotopy groups . We also show that has a central series whose successive quotients are , which are direct sums of homotopy groups of p-local spheres. This leads to a determination of the order of the p-torsion subgroup of and an upper bound for its p-exponent. These results apply to any Lie group G at a regular prime p. We derive some general properties of and give numerous explicit calculations. Received: 14 April 2001; in final form: 10 September 2001 / Published online: 17 June 2002     

Bordism-Finiteness and Semi-simple Group Actions

We give bordism-finiteness results for smooth S ³-manifolds. Consider the class of oriented manifolds which admit an S ¹-action with isolated fixed points such that the action extends to an S ³-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontryagin class p ₁, for example p ₁ = 0, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with S ³-action as above.     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Four-manifolds with π<Subscript>1</Subscript>–free second homotopy

 We study the homotopy type of closed connected orientable topological 4–manifolds M with Λ–free second homotopy group, where Λ is the integral group ring of π₁(M). This is related with problem N.4.53 of [23], and extends some results proved for the class of closed 4–manifolds with free fundamental group [10][11]. Other applications on special classes of closed topological manifolds complete the paper. Received: 27 November 2001 / Revised version: 28 October 2002 Published online: 19 May 2003 Work performed under the auspices of the GNSAGA of the CNR of Italy and partially supported by Ministero per la Ricerca Scientifica e Tecnologica of Italy within the project Proprietà Geometriche delle Varietà Reali e Complesse, and by a research grant of the University of Modena and Reggio Emilia. Mathematics Subject Classification (2000): Primary: 57 N 65, 57 R 67; Secondary: 57 Q 10, 57 R 80     

Genericity of exponential dichotomy for two-dimensional differential systems

We consider the setD of two-dimensional, quasi-periodic, traceless linear systems with a fixed number k>-2 of frequencies, which exhibit an exponential dichotomy. We show thatD is generic in the C⁰-topology in the set of all such two-dimensional linear systems. Entrata in Redazione l'11 dicembre 1998. Key Words: Exponential dichtomy, rotation number, quasi-periodic linear systems. AMS (MOS) Subject Identification Numbers: 34 D 05, 54 H 20, 58 F 15. R. Fabbri was supported by the Doctoral Program in Mathematics and by a research grant of the University of Florence. R. Johnson was supported by the G.N.A.F.A. and the M.U.R.S.T.     

Classification up to cobordism of manifolds with simple action of ℤ/p

We deal with quasi-complex manifolds with an action of the group ℤ/p such that the set of fixed points of this action has a trivial normal bundle. The set of cobordism classes of these manifolds is described in terms of the coefficients of the formal group of geometric cobordisms and in terms of characteristic numbers. We also establish the relationship between this work and relevant papers containing a solution of this problem in some particular cases. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 260–268, February, 1998. The author is greatly indebted to his advisor Professor V. M. Bukhshtaber for his active interest in this work and useful discussions. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01404.     

R-nilpotency in homotopy equivalences

We study the monoid of self homotopy equivalences of anR-nilpotent space, with the goal of understanding the actions of a cyclic group of orderp on a simply-connected homologically finite space with uniquelyp-divisible homotopy groups. This work was supported in part by the National Science Foundation.     

Continuous group actions on profinite spaces

For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point spaces and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite étale topological types of varieties over a field. One motivation is to understand Grothendieck’s section conjecture in terms of homotopy fixed points.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).