Separation by a codimension-1 map with a normal crossing point

Letf:M ^n–1N ⁿ be an immersion with normal crossings of a closed orientable (n–1)-manifold into an orientablen-manifold. We show, under a certain homological condition, that iff has a multiple point of multiplicitym, then the number of connected components ofN–f(M) is greater than or equal tom+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math. 36 (1992), 500–504) and Biasi, Motta and Saeki (Topology Appl. 52 (1993), 81–87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicitym. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.     

Dimensions of fixed point sets of involutions

Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fⁿ and F² of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fⁿ is small if the normal bundle of F² does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that . In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F² in M; further, we determine in these cases the equivariant cobordism class of (M, T). Received: 25 August 2005     

Rohlin’s invariant and gauge theory,I. Homology 3-tori

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H ¹ (Y;₂) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Cassons homology sphere invariant reduces mod 2 to the Rohlin invariant.     

Symmetric group actions on homotopy S 2 × S 2

Let X be a closed smooth 4-manifold which is homotopy equivalent to S ² × S ². In this paper we use Seiberg-Witten theory to prove that if X admits a spin symmetric group S ₄ action of even type with b ₂ ⁺ (X/S ₄) = b ₂ ⁺ (X), then as an element of R (S ₄), Ind _S4 D _X = k ₁ (1 − θ) + k ₂(ψ₁ − ψ₂) for some integers k ₁ and k ₂, where 1, θ, ψ₁, ψ₂ are irreducible characters of S ₄ of degree 1, 1, 3, and 3 respectively. Authors’ address: Ximin Liu and Hongxia Li, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China     

Path coupling without contraction

Path coupling is a useful technique for simplifying the analysis of a coupling of a Markov chain. Rather than defining and analysing the coupling on every pair in Ω×Ω, where Ω is the state space of the Markov chain, analysis is done on a smaller set SΩ×Ω. If the coefficient of contraction β is strictly less than one, no further analysis is needed in order to show rapid mixing. However, if β=1 then analysis (of the variance) is still required for all pairs in Ω×Ω. In this paper we present a new approach which shows rapid mixing in the case β=1 with a further condition which only needs to be checked for pairs in S, greatly simplifying the work involved. We also present a technique applicable when β=1 and our condition is not met.     

Topology and the robot arm

A robot arm is in effect a smooth function from the space of positions of the arm to the space of positions of a coordinate frame attached to the end of the arm. For the most common robots built today, this means a map f: T ⁿ R ³×SO ₃. We describe the singularities of this map. The set of rotational singularities is the set of arm positions where the axes of the links are parallel to a plane. Thus, it is always two-dimensional. Also, we show that f is homotopic to a map which factors through a circle, and represents the generator of ₁(SO ₃). The engineering implication of these statements are discussed.     

Euler homology

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring of a topological space X. This homology theory Eh _* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh _*, generalizing the equivariant Euler characteristic.     

Exceptional Extended Dual Polar Spaces

We address the classification problem of flag-transitive geometries with diagrams of the form where the leftmost edge symbolizes the geometry of vertices and edges of a complete graph ons + 2 vertices and the residue of an element of the leftmost type is a finite thick classical dual polar space. These geometries are known asextended dual polar spaces. An extended dual polar space is called affine if it possesses a flag-transitive automorphism group which contains a normal subgroup acting regularly on the set of elements of the leftmost type. For a dual polar space D with three points per line there exists a unique 2-simply connected affine extension A(D) of D. We show that a flag-transitive extended dual polar space is either a quotient of A(D) for some D or isomorphic to one of 19 exceptional geometries whose full automorphism groups are isomorphic respectively to Sym₈,U₄(2).2,Sp₆(2) × 2,Sp₆(2), 3 · U4(3).2²,U₄(3).2²,U₅(2).2,McL.2,HS.2,Suz.2,Sp₈(2), 3 · Fi₂₂.2,Fi₂₂.2,Co₂ × 2,Co₂,Fi₂₄(s = 4,t = 2),Fi₂₄(s = t = 3),F₁andFi₂₃.     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.     

Transformations preserving normality and Wishart-ness

Let R^n×p, (n), Gl(p) and ₊(p) denote respectively the set of n×p matrices, the set of n×n orthogonal matrices, the set of p×p nonsingular matrices and the set of p × p positive definite matrices. In this paper, it is first shown that a bijective and bimeasurable transformation (BBT) g on R^p≡R^p×1 preserving the multivariate normality of N_p(μ, Σ) for fixed μ=μ₁, μ₂ (μ₁≠μ₂) and for all Σ ₊(p) is of the form g(x)=Ax+b a.e. for some (A, b)Gl(p)×R^p. Second, a BBT g on R^n×p preserving the form for certain 's and all Σ ₊(p) is shown to be of the form g(x)=QxA+E a.e. for some (Q, A, E) (n)×Gl(p)×R^n×p. Third, a BBT h on ₊(p) preserving the Wishart-ness of W_p(Σ, m) (m≥p) for all Σ ₊(p) is shown to be of the form h(w)=A′wA a.e. for some AGl(p). Fourth, a BBT k(x, w)=(k₁(x, w), k₂(x, w)) on R^n×p× ₊(p) which preserves the form of for certain 's and all Σ ₊(p) is shown to be of the form k(x, w)=(QxA+E, A′wA) a.e. for some (Q, A, E) (n)×Gl(p)×R^n×p.     

Une famille remarquable de suites recurrentes lineaires

Letu(n) be a recurrent sequence of rational integers, i.e.,u(n+s)+a _s–1 u(n+s–1)+...+a ₀ u(n)=0,n0,a _i,i=0,...,s–1. The polynomialP(x)=x ^s +a _s–1x^s +...+a ₀ is the companion or the characteristic polynomial of the recurrence. It is known that if none of the ratios of the roots ofP is a root of unity, then the setA={n,u(n)=0} is finite. A recent result of F. Beukers shows that ifs=3, then the setA has at most 6 elements and there exists, up to trivial transformations, only one recurrence of order 3 with 6 zeros, found by J. Berstel. In this paper, we construct for eachs, s2 a recurrent sequence of orders, with at leasts ²/2+s/2–1 zeroes, which generalize Berstel's sequence.

     

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

A primary obstruction to topological embeddings¶and its applications

For a proper continuous map f:M→N between topological manifolds M and N with m≡ dimM < dimN≡m+k, a primary obstruction to topological embeddings θ(f) ∈H ^c _{m − k} (M; Z ₂) has been defined and studied by the authors in {9, 8, 2, 3], where H ^c _* denotes the singular homology with closed support. In this paper, we study the obstruction from the viewpoint of differential topology and give various applications. We first give some characterizations of embeddings among generic differentiable maps, which are refinements of the results in [9, 10]. Then we give a result concerning the number of connected components of the complement of the image of a codimension-1 continuous map with a normal crossing point, which generalizes the results in [6, 4, 5, 9]. Finally we give a simple proof of a theorem of Li and Peterson [20] about immersions of m-manifolds into (2m-1)-manifolds. Received: 3 December 1999 / Revised version: 10 October 2000     

Symplectic tori in rational elliptic surfaces

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive 2-dimensional homology class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface . We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.     

Shuffling algorithm for boxed plane partitions

We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from a×b×c to (a−1)×(b+1)×c or (a+1)×(b−1)×c. Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions.Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties.     

Geometric formulas for Smale invariants of codimension two immersions

We give three formulas expressing the Smale invariant of an immersion f of a (4k−1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k−1)-space.The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k−1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least a_k(2k−1)! cusps, where a_k=2 if k is odd and a_k=1 if k is even.     

Circle bundles over 4-manifolds

Every 1-connected topological 4-manifold M admits a S¹-covering by # _{r − 1} S² × S³, where Received: 4 July 2004     

Robust estimation in the linear model with asymmetric error distributions

In the linear model X^{n × 1} = C^{n × p}θ^{p × 1} + E^{n × 1}, Huber's theory of robust estimation of the regression vector θ^{p × 1} is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector E^{n × 1}. In the first model considered, the restriction of F to a set [−a₀, b₀] is a standard normal distribution contaminated, with probability , by an unknown distribution symmetric about 0. In the second model, the restriction of F to [−a₀, b₀] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [−a₀, b₀] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θ^{p × 1} are constructed, under mild regularity conditions on the sequence of design matrices {C^{n × p}}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].     

A generalized Conner-Floyd conjecture and the immersion problem for low 2-torsion lens spaces

Let α(d) denote the number of ones in the binary expansion of d. For 1?k?α(d) we prove that the 2(d+α(d)−k)+1-dimensional, 2^k-torsion lens space does not immerse in a Euclidian space of dimension 4d−2α(d) provided certain technical condition holds. The extra hypothesis is easily eliminated in the case k=1 recovering Davis’ strong non-immersion theorem for real projective spaces. For k>1 this is a deeper problem (solved only in part) that requires a close analysis of the interaction between the Brown-Peterson 2-series and its 2^k analogue. The methods are based on a partial generalization of the Brown-Peterson version for the Conner-Floyd conjecture used in this context to detect obstructions for the existence of Euclidian immersions.