Classification of immersions of graphs into a plane

In 2005, R. Nikkuni calculated the Wu invariant for immersions of graphs into a plane considered up to a regular homotopy, i.e., for homotopies that are immersions. He showed that two immersions are regularly homotopic if and only if their Wu invariants coincide. In this paper a simple combinatorial construction for this invariant is described, a theorem similar to Ryo Nikkuni’s theorem is proved, and the fact that all values of the constructed combinatorial invariant can be implemented by immersions is also proved.     

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.     

Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants

In this paper, a geometric approach to stable homotopy groups of spheres based on the Pontryagin–Thom construction is proposed. From this approach, a new proof of the Hopf-invariant-one theorem of J. F. Adams for all dimensions except 15, 31, 63, and 127 is obtained. It is proved that for n > 127, in the stable homotopy group of spheres Π _n , there is no element with Hopf invariant one. The new proof is based on geometric topology methods. The Pontryagin–Thom theorem (in the form proposed by R. Wells) about the representation of stable homotopy groups of the real, projective, infinite-dimensional space (these groups are mapped onto 2-components of stable homotopy groups of spheres by the Kahn–Priddy theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, nonoriented) is considered. The Hopf invariant is expressed as a characteristic class of the dihedral group for the self-intersection manifold of an immersed codimension-1 manifold that represents the given element in the stable homotopy group. In the new proof, the geometric control principle (by M. Gromov) for immersions in the given regular homotopy classes based on the Smale–Hirsch immersion theorem is required. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 3–15, 2007.     

Cayley射影平面的浸入Kaehler曲面

得到了Cayley射影平面的全复浸入Kaehler曲面的一些有趣的拓扑限制。     

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     

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory.     

Finite order q-invariants of immersions of surfaces into 3-space

Given a surface F, we are interested in valued invariants of immersions of F into , which are constant on each connected component of the complement of the quadruple point discriminant in . Such invariants will be called “q-invariants.” Given a regular homotopy class , we denote by the space of all q-invariants on A of order . We show that ifF is orientable, then for each regular homotopy class A and each n, $\dim (V_n (A) / V_{n-1}(A) ) \leq 1$. Received June 15, 1999; in final form September 22, 1999 / Published online October 30, 2000     

Unique embeddings of simple projective plane polyhedral maps

We characterize the 3-valent polyhedral maps in the projective plane whose graphs have a unique embedding in the projective plane. This is done by demonstrating two forbidden subgraphs of the dual of these uniquely embeddable graphs.     

A steinitz-type theorem for the projective plane

A complete characterization is given for those graphs which are the 1-skeletons of cell-decompositions of the projective plane. The characterization uses properties of the edge-graph of a centrally symmetric 3-polytope and it may serve as a tool for investigations on Hamiltonian circuits in the projective plane.     

Homotopy classification of projective convex sets

A subset of projective space is called convex if its intersection with every line is connected. The complement of a projective convex set is again convex. We prove that for any projective convex set there exists a pair of complementary projective subspaces, one contained in the convex set and the other in its complement. This yields their classification up to homotopy.     

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber on the deleted product criteria for embeddings and immersions. The isotopy and regular homotopy versions of the above theorems are also improved. We show by examples that they cannot be improved further. These results have many interesting corollaries, e.g.? 1) Any closed homologically 2-connected smooth 7-manifold smoothly embeds in .? 2) If and then the set of PL embeddings up to PL isotopy is in 1-1 correspondence with?. Received: July 6, 2000     

A characterization of constant isotropic immersions by circles

Motivated by the well-known result of Nomizu and Yano [4], we provide a characterization of constant isotropic immersions into an arbitrary Riemannian manifold by circles on the submanifolds. As an immediate consequence of this result, we characterize Veronese imbeddings of complex projective spaces into complex projective spaces which are typical examples of Kähler immersions. Received: 11 January 2002     

An extension of Calabi's rigidity theorem to complex submanifolds of indefinite complex space forms

The rigidity for full holomorphic isometric immersions of an indefinite Kähler manifold into an indefinite complex space form is proved. All such immersions between indefinite complex projective (and hyperbolic) spaces are founded and examples of non-congruents holomorphic isometric immersions are exposed.     

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

Annals of Global Analysis and Geometry - A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the...     

Bounding the number of embeddings of 5-connected projective-planar graphs

A graph is said to be projective-planar if it is nonplanar and is embeddable in a projective plane. In this paper we show that the numbers of projectiveplanar embeddings (up to equivalence) of all 5-connected graphs have an upper bound c(?120).     

ON DEGREES OF MINIMAL IMMERSIONS OF SYMMETRIC SPACES INTO SPHERES

This paper considers the degrees of minimal immersions of symmetric spaces intospheres.A practical way to count the degree of a given regular minimal immersion of acompact irreducible symmetric space into sphere S_1~n is given.As an application,degrees ofregular minimal immersions of all rank one compact symmetric spaces into spheres arecounted.     

Ten Exceptional Geometries from Trivalent Distance Regular Graphs

The ten distance regular graphs of valency 3 and girth > 4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.     

Cubic irreducible graphs for the projective plane

A characterization of all cubic finite graphs that do not embed in the real projective plane P is given in the sense that Kuratowski characterized all non-planar finite graphs. Specifically it is shown that there exist exactly 6 cubic irreducible graphs for P.     

On the spectral geometry for the Jacobi operators of harmonic maps into a quaternionic projective space

We characterize both invariant and totally real immersions into the quaternionic projective space by the spectra of the Jacobi operator. Also, we study spectral characterization of harmonic submersions when the target manifold is the quaternionic projective space.