Homotopy Classes Of Maps Research Articles

Least area incompressible surfaces in 3-manifolds

Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function. The existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group [18, 17]. In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly. One should note that evidence supporting this view had been found by Meeks-Yau in their embedding theorems for minimal disks and 2-spheres [13, 143 . Our main result asserts that if a least area immersion is homotopic to an embedding, then it has no self-intersections, which clearly exemplifies the above philosophy. The precise result is the following.

On the Segal Conjecture for Z 2 × Z 2

The Segal conjecture regarding the Burnside ring and stable cohomotopy of a finite group G is reduced for the case G = Z2X Z2 to a statement about Ext groups. This statement has since been proved by H. Miller, J. F. Adams and J. H. C. Gonawardena. The Segal conjecture states that for any finite group G, there is an isomorphism (G: A (G) -> 7T(BG) from the completed Burnside ring to the zeroth stable cohomotopy group of its classifying space. The conjecture was proved for cyclic groups in [4, 2, and 9]. In this paper we reduce the conjecture for Z2 X Z2 to a statement, (1), about Ext groups. This Ext statement was conjectured by Davis in [1] based upon extensive calculations and is proved by Adams, Gunawardena, and Miller in [7]. Let P = Z2[x, x'] be made into a module over the mod 2 Steenrod algebra A as in [5], and for S < k < n < xo let Pk' be the subquotient of P which is nonzero in degree k through n, inclusive. If n = x or k = -x, they will usually be omitted from the notation. The suspension liM of a graded module M is defined by (7Jm),+j= Mi. STATEMENT 1. There is an epimorphism of A -modules EP (g) Ep_ 1 Z2 (D EPO which induces an isomorphism in ExtA (, Z2). In fact p(sxa 0 sx b) = (?~I(?~~'(+ a+,)Sa+b+ I) +(SX () SX = ((a0+l)(b0+l) (a+ b+l)SX ) As mentioned above, Statement 1 is proved in [7]. The main result of this paper is THEOREM 2. Statement 1 implies that CZ2 XZ2 is an isomorphism. The proof of Theorem 2 mimicks [4]. The main part is to use Statement 1 to calculate 'u?(RP' x RP ) via the Adams spectral sequence. We begin by deducing from (1) the Ext groups relevant to [RPI A RP,I, S0I, the group of stable homotopy classes of maps. DEFINITION 3. If M is a (left) A-module, let DM denote the dual module, made into a left A-module using the antiautomorphism X; i.e., (DM)k = HomZ2(M k, Z2) with (9(+)(m) = ((X(O)m). Received by the editors January 30, 1981. 1980 Mathematics Subject Classification Primary 55Q10; Secondary 55T15.

Homotopy Classes of Maps to an Aspherical Complex

Homotopy Classes of Maps to an Aspherical Complex

Deformation of maps to branched coverings in dimension three

In this paper we address the problem of finding the simplest possible representative within each homotopy class of maps between 2-dimensional manifolds. We shall show that each such surface map of nonzero degree is homotopic to one of an especially simple form. To formulate the main result, define a map f: M -> N between surfaces to be a pinch if there is a compact, connected submanifold M1 c M, with boundary consisting of a single simple closed curve in the interior of M, such that N = M/M1, the quotient of M with Ml identified to a point, and such that f is the quotient map. Define a surjective mapf: M --> N between surfaces to be a branched covering if f is finite-to-one and there exists a finite set B c N, perhaps empty, such that f I f'-(N - B) is an ordinary covering space projection.

Self-intersections and higher Hopf invariants

IN THIS paper we show how the well known models for loop spaces of Boardman and Vogt [3], James [5], May [9], and Segal[ lo], can be viewed in a natural way as “Thorn spaces for immersions”. Thus homotopy classes of maps into these models correspond to bordism classes of immersed manifolds with certain extra structures. By considering the multiple points of such immersions we obtain operations in homotopy theory. Special cases are the generalised and higher Hopf invariants of James[6], the Hopf ladder of Boardman and Steer[2], and the cohomotopy operations of Snaith [ 121, and Segal[ I I]. In 01 we establish the connection between models of loop spaces and structured immersions. In 02 we describe the process of “taking k-tuple points” in what might be termed an “external Hopf invariant”, and set out its properties in Theorem 2.2. We then get the Segal and Snaith operations by composing with a suitable “forgetful” function. A similar procedure is followed in 03 where the James operations are described. We are grateful to the referee for many helpful comments.

The existence of minimal immersions of two-spheres

In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of isolated points, where the structure is that of a branch point. We obtain an existence theory for harmonic maps of orientable surfaces into Riemannian manifolds via a complete existence theory for a perturbed variational problem. Convergence of the critical maps of the perturbed problem is sufficient to produce at least one harmonic map of the sphere into the Riemannian manifold. A harmonic map from a sphere is in fact a conformal branched minimal immersion. We prove the existence of minimizing harmonic maps in two cases. If N is a compact Riemannian manifold with j2(N) = 0, then every homotopy class of maps from a closed orientable surface M to N contains a minimiz

Free 𝑆³-actions on 2-connected nine-manifolds

In this paper a classification of free S 3 {S^3} -actions on 2-connected 9-manifolds is obtained by examining the corresponding principal S 3 {S^3} -bundles. The orbit spaces that may occur are determined and it is proved that there are exactly two homotopy classes of maps from each of these spaces into the classifying space for principal S 3 {S^3} -bundles. It is shown that the total spaces of the corresponding bundles are distinct, yielding the main result that for each nonnegative integer k, there exist exactly two 2-connected 9-manifolds which admit free S 3 {S^3} -actions and, furthermore, the actions on each of these manifolds are unique.

Pull-Backs in Homotopy Theory

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no essential map between Eilenberg-MacLane spaces of different dimensions has a kernel. In this paper we define homotopy pull-backs and push-outs, which do exist and which behave like pull-backs and push-outs, and we give some of their properties. Applications may be found in [3; 5; 6 and 14].

Gottlieb Sets and Duality in Homotopy Theory

Evaluation subgroups of the homotopy groups have been objects of extensive study recently by Gottlieb, Haslam, Jerrold Siegel, G. E. Lang (Jr), etc. In [8] one of the authors has introduced the notions of ‘cyclic' and ‘cocyclic’ maps and studied generalizations of evaluation subgroups and their duals in the set up of Eckmann-Hilton duality. This paper continues the study of these generalized Gottlieb and dual Gottlieb subsets. All the spaces, except the function spaces, will be arc connected locally compact CW-complexes with base point at a vertex. For any X, Y the set of base point preserving homotopy classes of maps of X to Y is denoted by [X, Y].

Homotopy classes of maps between pro-spaces.

Homotopy classes of maps between pro-spaces.

The group of homotopy self-equivalences of some compactCW-complexes

For a space X, there is the problem of determining its group of homotopy self-equivalences ¢(X), i.e. those (based) homotopy classes of maps from X to itself which are invertible with respect to composition. One approach has been to look at the Postnikov decomposition of X: starting with the observation that for an Eilenberg-Mac Lane space 8(K(n, n))_~ Autrc, one can study g for induced fibrations and derive certain properties of 8(X). But there seems little hope to actually determine 8(X) in this way for spaces which have infinitely many non-zero homotopy groups, such as almost all compact spaces. In fact, the only results known to the author in this context are the homotopy-type classification of lensspaces (Franz, J.H.C. Whitehead, Olum) and the calculation of 8 for the pseudo-projective planes (Olum [3]). We describe here an approach which seems more suitable in the compact case. To any cellular map from one C W complex to another belongs its chain-homomorphism, i.e. the homomorphism induced on the (cellular) chaincomplexes. For chain-homomorphisms as well as for maps, there is the notion of homotopy, and a function is defined from homotopy classes to chain-homotopy classes. One can ask when this function is a one-to-one correspondence. Since our interest lies with non-simply connected spaces, it is better to look instead at the homomorphisms induced on the chaincomplexes of the universal coverings. We study a class of spaces for which the so-obtained function is a bijection; the class includes all compact complexes X such that nk(X)=0 for 1 <k < dimX. For such spaces g(X) and the group ~(X) of simple selfequivalences have purely algebraic descriptions. The paper starts with stating the basic classification theorem. Its proof is an application of the obstruction theory for non-simply connected spaces and is deferred to the end. It specializes to two quite explicit descriptions of the normal subgroup g~(X) of g(X) consisting of those self-equivalences which induce the identity on nl. We also discuss briefly

Homotopical Nilpotency of Loop-Spaces

In this paper we shall work in the category of countable CW-complexes with base point and base point preserving maps. All homotopies shall also respect base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class. Given spaces X, Y, we denote the set of homotopy classes of maps from X to Y by [X, Y]. We have an isomorphism τ: [∑X, Y] → [X, Ω Y] taking each map to its adjoint, where ∑ is the suspension functor and Ω is the loop functor. We shall denote τ(1 ∑x) by e′ and τ-1(1Ωx) by e.

Some properties of homotopy classes of maps of Kan spectra

The aim of this note is the study of the relations between the homotopy classes of maps of a spectrum A into a Kan spectrum B and the homotopy classes of maps of the components of the associated prespectrum A k into the spectrum B. The result is a little more general and consists in proving the validity for the spectra of MILNOR'S theorem, i.e. that for any countable family of subspectra A~ of A for which UA~=A and A~_A~+ ct

Maps which Induce the Zero Map on Homotopy

In this paper, all spaces will have the homotopy type of simply connected CW-complexes, and will have base points which are preserved by maps and homotopies. We denote by [X, Y] the set of homotopy classes of maps from X to Y, and by N[X, Y] the subset of those homotopy classes [ƒ] which induce the zero homomorphism on homotopy, that is, is the zero homomorphism for each i.

Simplicial and Homotopical Cohomology of Polyhedra

It is well known that, on the category of finite polyhedra, any two cohomology theories, satisfying the Eilenberg-Steenrod axioms, are isomorphic. Examples of such theories are simplicial cohomology and homotopical cohomology (the latter is defined by means of homotopy classes of maps into Eilenberg-MacLane spaces). In the case of polyhedra, using triple sequences and spectral sequences, one obtains a deep insight into the relationship between general cohomology theories (without the dimension axiom) and ordinary simplicial cohomology (1, p. 66). As a corollary the abovementioned uniqueness of cohomology theories satisfying the dimension axiom is obtained.

Representatives of homotopy classes of mappings into spheres

Representatives of homotopy classes of mappings into spheres