Topological stability through extremely tame retractions

Suppose that F:(Rn×Rd,0)→(Rp×Rd,0) is a smoothly stable, Rd-level preserving germ which unfolds f:(Rn,0)→(Rp,0); then f is smoothly stable if and only if we can find a pair of smooth retractions r:(Rn+d,0)→(Rn,0) and s:(Rp+d,0)→(Rp,0) such that f°r=s°F. Unfortunately, we do not know whether f will be topologically stable if we can find a pair of continuous retractions r and s.The class of extremely tame (E-tame) retractions, introduced by du Plessis and Wall, are defined by their nice geometric properties, which are sufficient to ensure that f is topologically stable.In this article, we present the E-tame retractions and their relation with topological stability, survey recent results by the author concerning their construction, and illustrate the use of our techniques by constructing E-tame retractions for certain germs belonging to the E- and Z-series of singularities.     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Topological structure of complete Riemannian manifolds with cyclic holonomy groups

Let M be a complete m-dimensional Riemannian manifold with cyclic holonomy group, let X be a closed flat manifold homotopy equivalent to M, and let L→X be a nontrivial line bundle over X whose total space is a flat manifold with cyclic holonomy group. We prove that either M is diffeomorphic to X×Rm-dimX or M is diffeomorphic to L×Rm-dimX−1.     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Homeomorphism with zero Jacobian: Sharp integrability of the derivative

Let N?2. We construct a homeomorphism f∈W1,1(N[0,1],RN) such that Jf=0 almost everywhere and sup0<ε?N−1εN[0,1]_∫|Df|^N−ε<∞. In particular, f∈W1,p(N[0,1],N[0,1]) for all p∈[1,N).     

High distance Heegaard splittings from involutions

Fixed an oriented handlebody H=H₊ with boundary F, let η(H₊)=H₋ be the mirror image of H₊ along F, so η(F) is the boundary of H₋, for a map f:F→F, we have a 3-manifold by gluing H₊ and H₋ along F with attaching map f, and denote it by Mf=H_+∪f:F→FH₋. In this note, we show that there are involutions f:F→F which are also reducible, such that Mf have arbitrarily high Heegaard distances.     

On the total absolute curvature of an immersed sphere

A recent result of Tobias Ekholm [T. Ekholm, Regular homotopy and total curvature II: Sphere immersions into 3-space, Alg. Geom. Topol. 6 (2006) 493-513] shows that for every ?>0 it is possible to construct a sphere eversion such that the total absolute curvature of the immersed spheres are always less than 8π+?. It is an open question whether this is the best possible. The paper contains results relating to this conjecture. As an interesting consequence of these methods it is shown that if during an eversion the total absolute curvature does not exceed 12π then a certain topological event must take place, namely the immersion must become non-simple at some point. An immersion f in general position is simple if for any irreducible self-intersection curve of f in 3-space, its two pre-image curves in the sphere are disjoint.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Non-simply connected copes

Let (W⁴,?W⁴) be a 4-manifold. Let f₁,f₂,…,f_k:(D²,?D²)→ (W⁴,?W⁴) be transverse immersions that have spherical duals $\text{α}{}_1\text{,α}{}_2\text{,…,α}{}_{\mathrm{k}}\text{:S}{}^2\text{→}\text{W}\text{?}$. Then there are open disjoint subsets V₁, V₂,…,V_k of W, such that for each 1?i?k, (a) ?V_i=V₁∩?W and ?V_i is an open regular neighborhood of f_i(?D²) in ?W, and (b) (V_i,?V_i,f_i(?D²)) is proper homotopy equivalent to (M, ?M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.     

Existence of infinitely many mountain pass solutions for some fourth-order boundary value problems with a parameter

In this paper, for the fourth-order boundary value problem (BVP) ,0<t<1,u(0)=u(1)=u^″(0)=u^″(1)=0, where f:[0,1]×R→R is continuous, η≤0 is a parameter, the existence of infinitely many mountain pass solutions are obtained with the variational methods and critical point theory. We prove the conclusion by combining sub-sup solution method, Mountain pass theorem in order intervals, Leray-Schauder degree theory and Morse theory.     

On multifractality and time subordination for continuous functions

Let Z:[0,1]→R be a continuous function. This paper relates to the existence of a decomposition of Z as Z=g○f, where g:[0,1]→R is a monofractal function with exponent 0<H<1 and f:[0,1]→[0,1] is a time subordinator, i.e. the integral of a positive Borel measure supported by [0,1]. An equivalent question consists of searching for a (multifractal) parametrization of Z which transforms Z into a monofractal function. We establish that such a decomposition can be found for a large class of functions which includes the usual examples of multifractal functions.We find an interesting relationship between self-similar functions and self-similar measures as an application of our results.Our theorems yield new insights in the understanding of the multifractal behaviour of functions, giving a significant role to the regularity analysis of Borel measures.     

Homotopy decomposition of a group of symplectomorphisms of S×S

We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of S²×S² when the ratio of the area of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.     

Smoothness of the images of members of a linear system under an endomorphism of the affine plane

Let φ=(f,g) be an endomorphism of the affine plane C² defined by two polynomials f,g∈C[x,y] and let Λ={C_b∣b∈C} be the pencil of lines C_b defined by x=b. We shall consider the smoothness criterion of the image curve φ(C_b). The hypersurface V whose coordinate ring is C[x,f,g] and the normalization of V will play interesting roles in analyzing the properties of the set φ(Λ)={φ(C_b)∣b∈C}.     

A fixed point theorem for the pseudo-circle

Let f:C→C be a self-map of the pseudo-circle C. Suppose that C is embedded into an annulus A, so that it separates the two components of the boundary of A. Let F:A→A be an extension of f to A (i.e. F|C=f). If F is of degree d then f has at least |d−1| fixed points. This result generalizes to all plane separating circle-like continua.     

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u)=−u^″+f(u) defined on a Sobolev space of periodic functions Hp(S¹), p?1. Let be the plane z=0 and, for n>0, let n be the cone x²+y²=tan²z, |z−2πn|<π/2; also set . For a generic smooth nonlinearity f:R→R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S¹),C) and (R³,Σ)×H where H is a real separable infinite-dimensional Hilbert space.     

Singular fibers and characteristic classes

Let be a proper generic map between smooth manifolds with dimN−dimM=−1. We explicitly calculate the cohomology class dual to the closure of the set of points in N over which lies a specific singular fiber in terms of characteristic classes of M and N.     

On the Whitehead group of Novikov rings associated to irrational homomorphisms

Given a homomorphism ξ:G→R we show that the natural map from the Whitehead group of G to the Whitehead group of the Novikov ring is surjective. The group is of interest for the simple chain homotopy type of the Novikov complex. It also contains the Latour obstruction for the existence of a nonsingular closed 1-form within a fixed cohomology class ξ∈H¹(M;R), where M is a closed connected smooth manifold.     

Good distance graphs and the geometry of matrices

Denote by G=(V,∼) a graph which V is the vertex set and ∼ is an adjacency relation on a subset of V×V. In this paper, the good distance graph is defined. Let (V,∼) and (V^′,∼^′) be two good distance graphs, and φ:V→V^′ be a map. The following theorem is proved: φ is a graph isomorphism ⇔φ is a bounded distance preserving surjective map in both directions ⇔φ is a distance k preserving surjective map in both directions (where k<diam(G)/2 is a positive integer), etc. Let D be a division ring with an involution such that both |F∩Z_D|?3 and D is not a field of characteristic 2 with D=F, where and Z_D is the center of D. Let H_n(n?2) be the set of n×n Hermitian matrices over D. It is proved that (H_n,∼) is a good distance graph, where A∼B⇔rank(A-B)=1 for all A,B∈H_n.