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.     

Immersions of the projective plane with one triple point

We consider C^∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C^∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.     

Generators for the tautological algebra of the moduli space of curves

In this paper, we prove that the tautological algebra in cohomology of the moduli space M_g of smooth projective curves of genus g is generated by the first [g/3] Mumford-Morita-Miller classes. This solves a part of Faber's conjecture (Moduli of Curves and Abelian Varieties Vieweg, Braunschweig, 1999) concerning the structure of the tautological algebra affirmatively. More precisely, for any k we express the kth Mumford-Morita-Miller class e_k as an explicit polynomial in the lower classes for all genera g=3k−1,3k−2,…,2.     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) fQ_| has 2g+2 critical points p₀,p₁,…,p2g+1 with f(p₀)<f(p₁)<?<f(p2g+1) where p₀ is a minimum and p2g+1 is a maximum, and p₁,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri<f(pi), then f−1(ri)∩Q consists of a circle if i is odd, and f−1(ri)∩Q consists of two circles if i is even.     

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.     

Nielsen periodic point theory for periodic maps on orientable surfaces

Let denote a periodic self map of minimal period m on the orientable surface of genus g with g>1. We study the calculation of the Nielsen periodic numbers NPn(f) and NΦn(f). Unlike the general situation of arbitrary maps on such surfaces, strong geometric results of Jiang and Guo allow for straightforward calculations when n≠m. However, determining NPm(f) involves some surprises. Because fm=idFg, fm has one Nielsen class Em. This class is essential because L(idFg)=χ(Fg)=2−2g≠0. If there exists k<m with L(fk)≠0 then Em reduces to the essential fixed points of fk. There are maps g (we call them minLef maps) for which L(gk)=0 for all k<m. We show that the period of any minLef map must always divide 2g−2. We prove that for such maps Em reduces algebraically iff it reduces geometrically. This result eliminates one of the most difficult problems in calculating the Nielsen periodic point numbers and gives a complete trichotomy (non-minLef, reducible minLef, and irreducible minLef) for periodic maps on Fg.We prove that reducible minLef maps must have even period. For each of the three types of periodic maps we exhibit an example f and calculate both NPn(f) and NΦn(f) for all n. The example of an irreducible minLef map is on F₄ and is of maximal period 6. The example of a non-minLef map is on F₂ and has maximal period 12 on F₂. It is defined geometrically by Wang, and we provide the induced homomorphism and analyze it. The example of an irreducible minLef map is a map of period 6 on F₄ defined by Yang. No algebraic analysis is necessary to prove that this last example is an irreducible minLef map. We explore the algebra involved because it is intriguing in its own right. The examples of reducible minLef maps are simple inversions, which can be applied to any Fg. Using these examples we disprove the conjecture from the conclusion of our previous paper.     

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     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

Normal families and value distribution in connection with composite functions

We prove a value distribution result which has several interesting corollaries. Let k∈N, let α∈C and let f be a transcendental entire function with order less than 1/2. Then for every nonconstant entire function g, we have that (f○g)^(k)−α has infinitely many zeros. This result also holds when k=1, for every transcendental entire function g. We also prove the following result for normal families. Let k∈N, let f be a transcendental entire function with ρ(f)<1/k, and let a₀,…,ak−1,a be analytic functions in a domain Ω. Then the family of analytic functions g such that

     

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S³ has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S³. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.     

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.     

On the Betti number of the image of a generic map

Let be a differentiable map of a closed m-dimensional manifold into an (m + k)-dimensional manifold with k > 0. We show, assuming that f is generic in a certain sense, that f is an embedding if and only if the (m - k + 1)-th Betti numbers with respect to the Čech homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings [BS1]. As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of [BR, BMS1, BMS2, Sae1] for immersions with normal crossings. Received: January 3, 1996     

An improvement of the Nevanlinna-Gundersen theorem

A well-known result of Nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane C and for four distinct values aj∈C∪{∞}, if νf−aj=νg−aj for all 1?j?4, then g is a Möbius transformation of f. In 1983, Gundersen generalized the result of Nevanlinna to the case where the above condition is replaced by: min{νf−aj,1}=min{νg−aj,1} for j=1,2 and νf−aj=νg−aj for j=3,4. In this paper, we prove that the theorem of Gundersen remains valid to the case where min{νf−aj,1}=min{νg−aj,1} for j=1,2, and min{νf−aj,2}=min{νg−aj,2} for j=3,4. Furthermore, we work on the case where {aj} are small functions.     

Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R³, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T⊕P⊕Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P,Q are Z/2 valued invariants characterized by the property that for any regularly homotopic immersions , P(i)−P(j)∈Z/2 (respectively, Q(i)−Q(j)∈Z/2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j.For immersion and diffeomorphism such that i and i○h are regularly homotopic we show:

P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h_∗−Id)+ε(deth∗∗))mod2     

Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem.     

Taylor series and divisibility

LetA be an augmentedK-algebra; defineT:A →A ?k _kA byT(a)=1?a ?a?1,a ∈A. We prove, under some conditions, thatg is in the subalgebraK[f] ofA generated byf if and only ifT(g) is in the principal ideal generated byT(f) inA?k _kA. WhenA=K[[X]],T(f) is a multiple ofT(X) if and only iff belongs to the ringL obtained by localizingK[X] at (X).     

Fractional weak discrepancy and split semiorders

The fractional weak discrepancywd_F(P) of a poset P=(V,?) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative k for which there exists a function f:V→R satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5] and [7] on the range of wd_F for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0≤s−1≤r<2s.     

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: M ⁿ → Rⁿ⁺¹ be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of M ⁿ into R ^{n +} 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.     

Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori

The multiplier spectral curve of a conformal torus f : T ² → S ⁴ in the 4-sphere is essentially (Bohle et al., Conformal maps from a 2-torus to the 4-sphere. arXiv:0712.2311) given by all Darboux transforms of f. In the particular case when the conformal immersion is a Hamiltonian stationary torus ${f: T^2 \to\mathbb{R}^4}$ in Euclidean 4-space, the left normal N : M → S ² of f is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of N, as defined in Hitchin (J Differ Geom 31(3):627–710, 1990). We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.