Smooth Approximation of Orlicz-Sobolev Maps Between Manifolds

The problem of density of smooth functions in Orlicz-Sobolev spaces of maps between compact manifolds, without topological restrictions, is addressed. Our contribution refines earlier results in the literature via ad hoc Orlicz space techniques.     

The homological singularities of maps in trace spaces between manifolds

We deal with mappings defined between Riemannian manifolds that belong to a trace space of Sobolev functions. The homological singularities of any such map are represented by a current defined in terms of the boundary of its graph. Under suitable topological assumptions on the domain and target manifolds, we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps. Moreover, we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.This research was partially supported by National Science Foundation grant GP-7952X1.     

On smooth Chas-Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In [A.J. Baker, C. Ozel, Complex cobordism of Hilbert manifolds with some applications to flag varieties, Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [C. Ozel, On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in preparation], by using Quinn's Transversality Theorem [F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. 15 (1970) 213-222], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [M. Chas, D. Sullivan, String topology, math.GT/9911159, 1999], Chas and Sullivan described an intersection product on the homology of loop space LM. In [R.L. Cohen, J.D.S. Jones, A homotopy theoretic realization of string topology, math.GT/0107187, 2001], R. Cohen and J. Jones described a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. In this paper, we will extend this product on cobordism and bordism theories.     

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.     

Compactness in the fine topology

For reasonable spaces (including topological manifolds) X, Y, we characterize compact subsets of the space of continuous maps from X to Y, topologized with the fine (Whitney) C⁰-topology. In the case of smooth manifolds, we characterize also compact subsets of the space of C^r maps in the Whitney C^r topology.     

On the Regularity Theories for Harmonic Maps from Finsler Manifolds

The author studies the regularity of energy minimizing maps from Finsler manifolds to Riemannian manifolds.It is also shown that the energy minimizing maps are smooth,when the target manifolds have no ...     

Banach流形上映射度

本文讨论一种无限维流形上拓扑不变量，即具有可定向Ｆｒｅｄｈｏｌｍ结构的实Ｂａｎａｃｈ流形上　　Ｃｒ　　映射度．它是通常有限维流形上光滑映射度的一种自然推广．     

Classification of homotopy Wall's manifolds

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a Wall's manifold (defined as a mapping torus of a Dold manifold), introduced by Wall in his 1960 Annals paper on cobordism, have been done by determining: (1) the normal invariants of Wall's manifolds, (2) the surgery obstruction of a normal invariant and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy Wall's manifolds (to be made precise in the body of the paper). Consequently classification results of automorphisms (self homeomorphisms, and self PL-homeomorphisms) of Dold manifolds follow.     

Rigorous and accurate enclosure of invariant manifolds on surfaces

Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.     

Schrödinger flow of maps into symplectic manifolds

The definition of Schrödinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrödinger flow of maps intoS ², and that there exists a unique local smooth solution for the initial value problem of one-dimensional Schrödinger flow of maps into Kahler manifolds. In case the targets are Kähler manifolds with constant curvature, it is proved that one-dimensional Schrödinger flow admits a unique global smooth solution.     

Topological Classification of m-Fields on Two- and Three-Dimensional Manifolds with Boundary

We consider m-fields that are generalizations of the Morse–Smale vector fields for manifolds with boundary. We construct complete topological invariants of m-fields on surfaces and m-fields without closed trajectories on three-dimensional manifolds. We also prove criteria for the topological equivalence of m-fields.     

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

Equivariant embedding theorems and topological index maps

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally non-singular maps with respect to a cohomology theory and show that oriented normally non-singular maps induce wrong-way maps on the chosen cohomology theory. For K-oriented normally non-singular maps, we also get a functor to Kasparov's equivariant KK-theory. We interpret this functor as a topological index map.     

Equivariant cohomology distinguishes toric manifolds

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are weakly isomorphic. We also prove that quasitoric manifolds, which can be thought of as a topological counterpart to toric manifolds, are equivariantly homeomorphic if and only if their equivariant cohomology algebras are isomorphic.     

Sobolev spaces and harmonic maps between singular spaces

Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces. Received December 6, 1996 / Accepted March 4, 1997     

On 1-connected 8-manifolds with the Same Homology as S~3×S~5

In this article, we classify 1-connected 8-dimensional Poincaré complexes, topological manifolds and smooth manifolds whose integral homology groups are isomorphic to those of S~3× S~5. A topic related to a paper of Escher and Ziller is also discussed.