Abstract:We extend results about critically k-colorable graphs to choosability and paintability (list colorability and on-line list colorability). Using a strong version of Brooks' Theorem, we generalize Gallai's Theorem about the structure of the low-degree subgraph of critically k-colorable graphs, and introduce a more adequate lowest-degree subgraph. We prove lower bounds for the edge density of critical graphs, and generalize Heawood's Map-Coloring Theorem about graphs on higher surfaces to paintability. We also show that on a fixed given surface, there are only finitely many critically k-paintable/k-choosable/k-colorable graphs, if k >= 6. In this situation, we can determine in polynomial time k-paintability, k-choosability and k-colorability, by giving a polynomial time coloring strategy for "Mrs. Correct". Our generalizations of k-choosability theorems also concern the treatment of non-constant list sizes (non-constant k). Finally, we use a Ramsey-type lemma to deduce all 2-paintable, 2-choosable, critically 3-paintable and critically 3-choosable graphs, with respect to vertex deletion and to edge deletion. (C) 2012 Elsevier B.V. All rights reserved.

Abstract:We prove that if phi is a homogeneous harmonic map from a Riemann surface M into a complex Grassmann manifold G(k,n), then the maps of the harmonic sequences generated by phi are all homogeneous.

Abstract:We give a necessary and sufficient condition for the fundamental group homomorphism of a map between CW-complexes (manifolds) to induce partial homology equivalences. As applications, we obtain characterizations of fundamental groups of homology spheres and Moore manifolds. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.

Abstract:We study low-dimensional representations (rigidity problem) of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds. (C) 2014 Elsevier Inc. All rights reserved.

Abstract:In this paper, we classify holomorphic curves in Qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{doc...

Abstract:Some polynomials P with rational coefficients give rise to well defined maps between cyclic groups, Z(q) -> Z(r), x + qZ -> P(x) + rZ. More generally, there are polynomials in several variables with tuples of rational numbers as coefficients that induce maps between commutative groups. We characterize the polynomials with this property, and classify all maps between two given finite commutative groups that arise in this way. We also provide interpolation formulas and a Taylor-type theorem for the calculation of polynomials that describe given maps. (C) 2014 Elsevier Inc. All rights reserved.

Abstract:In this paper we study the equivariant totally real immersions from S-3 into CPn. We first reduce these immersions to a system of algebraic equations by the unitary representations of SU(2). We give some explicit examples of minimal totally real isometric immersions from S-3/m(m+2)(3) into CPn, and characterize the minimal totally real isometric immersions from S-3/m(m+2)(3) into CPn by the standard example. We also give many minimal linearly full isometric immersions from S-1/5(3) into CP7. CP11 and CP15. As an application of our method, we classify equivariant Lagrangian S-3 in CP3 again. (C) 2012 Elsevier B.V. All rights reserved.

Abstract:In this paper we determine all homogeneous minimal immersions of 2-spheres in quaternionic projective spaces .

Abstract:In this article, we determine all homogeneous two-spheres in the complex Grassmann manifold G(2, 5; C) by theory of unitary representations of the 3-dimensional special unitary group SU(2).

Abstract:Let SLn(Z) for n >= 3 be the special linear group and M-r be a closed aspherical manifold. It is proved that when r < n, a group action of SLn (Z) on M-r by homeomorphisms is trivial if and only if the induced group homomorphism SLn(Z) -> Out(pi(1)(M)) is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. In particular, when pi(1)(M) is nilpotent, the group SLn(Z) cannot act nontrivially on M when r < n. This confirms a conjecture related to Zimmer's program for these manifolds.

Abstract:A homotopy theoretic description is given for trivial unit conjecture in the group ring ZG. (C) 2013 Elsevier Inc. All rights reserved.

Abstract:Let R be an infinite commutative ring with identity and n >= 2 an integer. We prove that for each integer i = 0, 1, ... , n - 2, the L-2-Betti number b(i)((2)) (G) vanishes when G is the general linear group GL(n)(R), the special linear group SLn(R) or the group E-n(R) generated by elementary matrices. When R is an infinite principal ideal domain, similar results are obtained when G is the symplectic group Sp(2n)(R), the elementary symplectic group ESp(2n)(R), the split orthogonal group O(n, n)(R) or the elementary orthogonal group EO(n, n)(R ). Furthermore, we prove that G is not acylindrically hyperbolic if n >= 4. We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of n-rigid rings.

Abstract:We obtain a sufficient and necessary condition for a finite group to act effectively on a closed flat manifold. Let G=En(R), EUn(R,?), SAut (Fn) or SOut (Fn). As applications, we prove that when n > 3 every group action of G on a closed flat manifold Mk (k