Extensions of codimension one immersions

We give a method for constructing all of the extensions of an immersion, and determine the CW structure and diffeomorphism type of each.

     

An intrinsic characterization of isometric pluriharmonic immersions with codimension one

We give an intrinsic characterization of isometric pluriharmonic immersions of Kähler manifolds into semi-Euclidean spaces with real codimension one, which is a generalization of the Ricci-Curbastro theorem.Research partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan.     

Rigidity of isometric immersions of higher codimension

Given an isometric immersionf:M ⁿ → ℝ ^N into Euclidean space, we provide sufficient conditions onf so that any 1-regular isometric immersion ofM ⁿ into ℝ ^N+1 is necessarily obtained as a composition off with a local isometric immersion ℝ ^N ⊃U → ℝ ^N+1 . This result has several applications.     

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.     

Thom polynomials, symmetries and incidences of singularities

As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials [T]. With the aid of this method we compute some sample results, such as the Thom polynomials associated to all stable singularities of codimension ≤8 between equal dimensional manifolds, and some other Thom polynomials associated to singularities of maps N ⁿ ?P ^n+k for k>0. We also give an application by reproving a weak form of the multiple point formulas of Herbert and Ronga ([H], [Ro2]). As a byproduct of the theory we define the incidence class of singularities, which — the author believes — may turn to be an interesting, useful and simple tool to study incidences of singularities. Oblatum 4-II-1999 & 19-VII-2000?Published online: 30 October 2000     

Equiaffine immersions of general codimension and its transversal volume element map

With an equiaffine immersion of codimension 1 into the affine space with the natural equiaffine structure, the conormal map is associated. In this paper, for an equiaffine immersion of general codimension into the space, we shall define the map corresponding to the conormal map, which is called the transversal volume element map. And we shall investigate if, an equiaffine immersion of general codimension into the space is determined by its affine fundamental form and its transversal volume element map.     

Schur and Schubert polynomials as Thom polynomials—cohomology of moduli spaces

The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new definition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems.     

Positivity of quiver coefficients through Thom polynomials

We use the Thom Polynomial theory developed by Fehér and Rimányi to prove the component formula for quiver varieties conjectured by Knutson, Miller, and Shimozono. This formula expresses the cohomology class of a quiver variety as a sum of products of Schubert polynomials indexed by minimal lace diagrams, and implies that the quiver coefficients of Buch and Fulton are non-negative. We also apply our methods to give a new proof of the component formula from the Gröbner degeneration of quiver varieties, and to give generating moves for the KMS-factorizations that form the index set in K-theoretic versions of the component formula.     

Ordinary subvarieties of codimension one

In this paper we extend the properties of ordinary points of curves [10] to ordinary closed points of one-dimensional affine reduced schemes and then to ordinary subvarieties of codimension one. Received: 20 March 1998 / Revised version: 7 July 1998     

Regular homotopy classes of immersions of 3-manifolds into 5-space

 We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and of the 3-torus T ³ with the following properties: (1) is regularly homotopic to F ₈ for some immersion , and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding. Received: 29 March 2001     

A geometric classification of immersions of 3-manifolds into 5-space

In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into ⁵ in a geometric manner. The pair (c(f), i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theory and extends the one defined in [10] for immersions with trivial normal bundle.Mathematics Subject Classification (2000): 57N35, 57R45, 57R42     

Topological invariants of stable immersions of oriented 3-manifolds in R4

We show that the $\mathbb{Z}$-module of first order local Vassiliev type invariants of stable immersions of oriented 3-manifolds into ${\mathbb{R}}^4$ is generated by 3 topological invariants: The number of pairs of quadruple points and the positive and negative linking invariants ${?}^{+}$ and ${?}^{?}$ introduced by V. Goryunov (1997) [7]. We obtain the expression for the Euler characteristic of the immersed 3-manifold in terms of these invariants. We also prove that the total number of connected components of the triple points curve is a non-local Vassiliev type invariant.     

Lie algebras of cohomological codimension one

We show that if is a finite dimensional real Lie algebra, then has cohomological dimension if and only if is a unimodular extension of the two-dimensional non-Abelian Lie algebra .