Embedding up to homotopy type— The first obstruction

For a 2n?m connected map from an n-dimensional complex to a m-dimensional manifold, an obstruction to embedding up to homotopy type is defined. The vanishing of this obstruction is a necessary and sufficient condition (in the 2n?m connected case, 2n?m ? 2, m?n ?3) to obtain an embedding up to homotopy type. In case the target manifold is Euclidean space, it is shown that the obstruction vanishes if and only if certain Thom operations are trivial. A classification theorem is given in the 2n?m+1 connected case.     

The geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable _boxclose/2{{\mathbb Z}/2} -equivariant map h(F) such that the stable \mathbb Z/2{{\mathbb Z}/2} -equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper, we express the geometric Hopf invariant of the Umkehr map F of an immersion f : M^m \looparrowright Nⁿ{f : M^m \looparrowright N^n} in terms of the double point set of f. We interpret the Smale–Hirsch–Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m <  2n – 1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.     

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

Compact abelian group extensions of discrete dynamical systems

Summary This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.     

Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

The classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA. In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C -analytic subspaces of R ^m. So, let X be a C -analytic subspace of R ^m and G a compact Lie group. The main result is a characterization of the real analytic G-principal fibre bundles over X for which the analytic and topological equivalence coincide. Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces. Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.     

Belts and κ-invariants of link maps in spheres

Linkings of higher dimensional disjoint singular spheres in euclidean space ℝ^m are compared with linkings in the sphereS ^m. Somewhat surprisingly, link homotopy inS ^m can be interpreted as a special case of link homotopy in ℝ^m. This leads to a considerable refinement of standard invariants such as the generalized Milnor invariants.     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is1+(m−1)(m−2)/2the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have?In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.?s as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.     

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551-1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space (X,ρ) can be extended to a triangular map F on X×I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let (X,f) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h(F)=h(f).     

On genus and cancellation in homotopy

Thegenus is determined for spaces of the homotopy type of aCW complex with one cell each in dimensions 0, 2n and 4n (and no other cells), such spaces providing the only cases of spaces with two non-trivial cells such that the homotopy class of the attaching map for the top cell is of infinite order and the genus of the space is non-trivial. The genus is characterised completely by two well understood invariants: theHopf invariant of the attaching map of the 4n-cell and the genus of thesuspension of the space. The algebraic tools are developed for the investigation of the ν-cancellation behaviour of these spaces and a cancellation theorem is proved: the homotopy type of a finite wedge of such spaces determines the homotopy type of each of the summands as long as the attaching maps of the 4n-cells all represent homotopy classes of infinite order. Comparing this result to known results aboutfinite co-H-spaces shows that the Hopf invariant is the single obstruction to such spaces admitting a co-H structure.     

Strong Homotopy Types,Nerves and Collapses

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial G-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of finite topological spaces.     

Rationalized evaluation subgroups of a map II: Quillen models and adjoint maps

We identify the long exact sequence induced on rational homotopy groups by the evaluation map , and in particular the rationalization of the evaluation subgroups of f, in terms of derivations of Quillen models and adjoint maps. We consider a generalization of a question of Gottlieb within the context of rational homotopy theory. We also study the rationalization of the G-sequence of a map. In a separate result of independent interest, we give an explicit Quillen minimal model of a product A×X, in the case in which A is a rational co-H-space.     

On phantom maps and a theorem of H. Miller

A mapf:X →Y is a phantom map if any composition off with a map from a finite complex intoX is null homotopic. The proof of the Sullivan conjecture by H. Miller enables us to understand more deeply this phenomena. We prove, among other things, that any map from a space with finitely many non-vanishing homotopy groups into a finite complex is phantom and that any fibration over a 2-connected space with finitely many non-vanishing homotopy groups and with fiber a finite complex is trivial over each skeleton of the base. Prof. Zabrodsky died in an automobile accident on November 20, 1986.     

Biharmonic map heat flow into manifolds of nonpositive curvature

Let M ^m and N be two compact Riemannian manifolds without boundary. We consider the L ² gradient flow for the energy . If and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 58E20, 58J35     

Complexes of Dirac operators in Clifford algebras

In this paper we study complexes of k Dirac operators (or variations of Dirac operators) in the real or complex Clifford algebras i.e. complexes in which the first map is induced by the matrix where is the Dirac operator with respect to the variable . In particular we prove that, if , the complex in the case of 3 operators can be described in terms of relations coming from the so called radial algebra. Moreover we show that if the dimension m is less than 2k-1, then the Fischer decomposition does not hold. Received: 2 February 2000; in final form: 20 June 2000 / Published online: 25 June 2001     

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.     

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X ₀ be a topological component of a nonsingular real algebraic variety and i:X → X _C is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X ₀ → X _C and obtain several results using rational homotopy theory and other standard tools of homotopy theory.     

Minimum numbers and Wecken theorems in topological coincidence theory. I

Minimum numbers measure the obstruction to removing coincidences of two given maps (between smooth manifolds M and N of dimensions m and n, resp.). In this paper, we compare them to four distinct types of Nielsen numbers. These agree with the classical Nielsen number when m = n (e.g., in the fixed point setting where M = N and one of the maps is the identity map). However, in higher codimensions, m − n > 0, their definitions and computations involve distinct aspects of differential topology and homotopy theory.