Benz planes of Dembowski type and groups of tangential projectivities

Using the tangential relation we introduce in Benz planes M of Dembowski type, which generalize the Benz planes over algebras of characteristic 2, the group ?? of tangential perspectivities. We prove that these groups have the same behaviour as the classical groups of projectivities if any tangential perspectivity is induced by an automorphism of M. As permutation groups of a circle onto itself the groups ?? essentially differs from the classical groups of projectivities. If M is a Laguerre plane of Dembowski type, then ?? is always sharply 3-transitive. For Minkowski planes of Dembowski type ?? is at least 2-transitive. If M is a finite Benz plane of order 2 ^s , then ?? is isomorphic to the group PGL ₂(2 ^s ) in its sharply 3-transitive representation.     

Constructing homotopy equivalences

In this paper we study the following construction of homotopy equivalences: Take a codimension one separating submanifold N^n?1 of Mⁿ, cut along N and glue the pieces together by a homeomorphism of N homotopic to the identity. Aside from the question of which homotopy equivalences can be so obtained, we will study qualitative questions such as stability, type of submanifold, etc. Relations to ΣΩ, the oozing problem in surgery theory, and Kervaire classes will be discussed.     

Rational homotopy of the (homotopy) fixed point sets of circle actions

In this paper, when G is the circle S¹ and M is a G-space, we study the rational homotopy type of the fixed point set MG, the homotopy fixed point set MhG, and the natural injection MG→MhG.     

Boundaries and complements of infinite- dimensional manifolds in the model space

Let M be a manifold modeled on a locally convex linear metric space E≌E^ω (or ≌E^ω_f and N a Z-submanifold of M. Then N is collared in M. In this paper, we study the following problem [1, 3]: Under what conditions can M be embedded in E so that N is the topological boundary of M in E? We gain a more mild sufficient condition than the previous papers [7, 8] and a necessary and sufficient condition in the case M has the homotopy type of Sⁿ (and each component of N is simply connected if n?2) and in the case N has the homotopy type of Sⁿ (n?2). Also we obtain a necessary and sufficient condition under which M can be embedded in E so that bd M = N and cl(E\M) has the homotopy type of Sⁿ (we assume that M and N are simply connected if n ? 2).     

Construction of nonseparable orthonormal compactly supported wavelet bases for L2（Rd）

Suppose M and N are two r × r and s × s dilation matrices,respectively.Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-TZr/Zr and N-TZs/Zs,respe...     

Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

One-basedness and groups of the form G/G 00

We initiate a geometric stability study of groups of the form G/G ⁰⁰, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M?? with a predicate for G ⁰⁰ and check 1-basedness or non-1-basedness for G/G ⁰⁰, where G is an additive truncation of M, a multiplicative truncation of M, SO ₂(M) or one of its truncations; such groups G/G ⁰⁰ are now interpretable in M??. We prove that the only 1-based groups are those where G is a sufficiently ??big?? multiplicative truncation, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-hyperdefinable groups constructed from those above.     

Erd?s-Zaks all divisor sets

Let ? _n be the finite cyclic group of order n and S ? ? _n . We examine the factorization properties of the Block Monoid B(? _n , S) when S is constructed using a method inspired by a 1990 paper of Erd?s and Zaks. For such a set S, we develop an algorithm in Section 2 to produce and order a set {M _i } _i=1 ^n?1 which contains all the non-primary irreducible Blocks (or atoms) of B(? _n , S). This construction yields a weakly half-factorial Block Monoid (see [9]). After developing some basic properties of the set {M _i } _i=1 ^n?1 , we examine in Section 3 the connection between these irreducible blocks and the Erd?s-Zaks notion of ??splittable sets.?? In particular, the Erd?s-Zaks notion of ??irreducible?? does not match the classic notion of ??irreducible?? for the commutative cancellative monoids B(? _n , S). We close in Sections 4 and 5 with a detailed discussion of the special properties of the blocks M1 with an emphasis on the case where the exponents of M ₁ take on extreme values. The work of Section 5 allows us to offer alternate arguments for two of the main results of the original paper by Erd?s and Zaks.     

Hyperbolic spatial graphs in 3-manifolds

For any closed connected orientable 3-manifold M, we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S³ via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M.     

Atomic Decomposition of Hardy Type Spaces on Certain Noncompact Manifolds

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X ^k (M), introduced in a previous paper of the authors, have an atomic characterization. An atom in X ^k (M) is an atom in the Hardy space H ¹(M) introduced by Carbonaro, Mauceri, and Meda, satisfying an ??infinite dimensional?? cancellation condition. As an application, we prove that the Riesz transforms of even order $\nabla^{2k} \mathcal{L}^{-k}$ map X ^k (M) into L ¹(T _2k M).     

HOMOTOPY SELF-EQUIVALENCE GROUPS OF UNIONS OF SPACES: INCLUDING MOORE-SPACES

Abstract

The group ?(M^m(A) v Mⁿ(π)) of homotopy self-equivalence classes of two Moore spaces is faithfully represented onto a (multiplicative) group of matrices for n≥m≥3. We consider, in this note, related representations of ?(M^m(Λ)vMⁿ(π)), for finitely generated Λ and π in the case where n≥4, and also where n=3 if ext(Λ, π)=0. The representation onto a matrix group, similar to that in the case above, is not, in general, valid. We show however that ?(M²(Λ)vMⁿ(π)) is represented onto ?(M²(Λ))× ?(Mⁿ(π) in this case, and that this representation determines an isomorphism with an iterated semi-direct product ?(M²(Λ)v Mⁿ(π)) ? {(Mⁿ(π), M²(Λ))? ext(π Λ ? π)} ? (?(M²(Λ)) × ? (Mⁿ(π)).

More generally we review, and-extend, the theory of the representation of the (generalized) near ring (XvY,XvY) onto the matrix (generalized) near-ring (XvY, XxY) where appropriate, in the case where X and Y are h-coloops; and we deduce results for the representation of ?(XvY, XvY). Some of the results published previously in the case of simply-connected CW co-h-spaces, extend to the case where X and Y are path-connected h-coloops one of which is well-pointed. We note the obstructions to the existence of a homomorphic section, and consider a number of special cases which occur when some of the groups are trivial.     

On R-robustly entropy-expansive diffeomorphisms

For a homoclinic class H(p _f ) of f ?? Diff¹(M), f?OH(p _f ) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set ?? _? (x) = {y ?? M: dist(g ⁿ (x), g ⁿ (y)) ?? g3 (?n ?? ?)} has zero topological entropy for each x ?? H(p _g ). We prove that there exists an open and dense set around f such that for every g in it, H(p _g ) admits a dominated splitting of the form E ?? F ₁ ?? ... ?? F _k ?? G where all of F _i are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown.     

Deformations of the hemisphere that increase scalar curvature

Consider a compact Riemannian manifold M of dimension n whose boundary ?M is totally geodesic and is isometric to the standard sphere S ^n?1. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n?1), then M is isometric to the hemisphere $S_{+}^{n}$ equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. In this paper, we construct counterexamples to Min-Oo??s Conjecture in dimension n??3.     

Short Geodesic Segments between Two Points on a Closed Riemannian Manifold

Let M ⁿ be a closed Riemannian manifold homotopy equivalent to the product of S ² and an arbitrary (n–2)-dimensional manifold. In this paper we prove that given an arbitrary pair of points on M ⁿ there exist at least k distinct geodesics of length at most 20k!d between these points for every positive integer k. Here d denotes the diameter of M ⁿ .     

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M ⁿ , g) be a compact Riemannian manifold with boundary ?M. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have ?M as a constant mean curvature hypersurface. We prove that this set is compact for dimensions n ?? 7 under the generic condition that the trace-free 2nd fundamental form of ?M is nonzero everywhere.     

Coquasitriangular Hopf group coalgebras and braided monoidal categories

Let ?? be a group, and let H be a Hopf ??-coalgebra. We first show that the category M ^H of right ??-comodules over H is a monoidal category and there is a monoidal endofunctor (F _?? , id, id) of M ^H for any ?? ?? ??. Then we give the definition of coquasitriangular Hopf ??-coalgebras. Finally, we show that H is a coquasitriangular Hopf ??-coalgebra if and only if M ^H is a braided monoidal category and (F _?? , id, id) is a braided monoidal endofunctor of M ^H for any ?? ?? ??.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

An index product formula for the study of elliptic resonance problems

Let $\text{(P)}\text{u}\text{?}\text{+ Au = f(u)}$ be a semilinear parabolic equation. If f(0) = 0 and f is of class C¹ in a neighborhood of 0, then there exists a local center manifold M near zero containing all small invariant sets of (P). The purpose of this paper is to prove an index product formula relating the homotopy index h(K) of a small isolated invariant set K relative to (P) to the homotopy index h_M(K) of the same set with respect to the equation induced by (P) on the center manifold M. This formula can be applied to elliptic BVP with resonance at zero. In particular, earlier results of Amann and Zehnder (Ann. Scuola Norm. Sup. Pisa IV7 (1980), 534–603) can be obtained under less restrictive assumptions than those used in that paper. Further-more, the formula permits applications to cases not discussed in Amann and Zehnder's paper. The applications of the index product formula are given in K. P. Rybakowski (Nontrivial solutions of elliptic boundary value problems with resonance at zero, Ann. Mat. Pura Appl., to appear).