Homotopy Classification Of Nyquist Maps

On minimal surfaces with free boundaries in given homotopy classes

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

In the present paper we construct an equivalence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related by this equivalence. This equivalence is given by a simple formula.

Detecting homotopy equivalences in base-point-free homotopy

Given a homotopy equivalence f : X → Y f:X \to Y , a homotopy inverse g of f, and a homotopy H : X × I → X H:X \times I \to X from g ∘ f g \circ f to 1 X {1_X} . We show that there is a homotopy K : Y × I → Y K:Y \times I \to Y from f ∘ g f \circ g to 1 Y {1_Y} such that f ∘ H ≃ K ∘ ( f × 1 I ) rel X × ∂ I f \circ H \simeq K \circ (f \times {1_I})\,{\text {rel}}\,X \times \partial I and H ∘ ( g × 1 I ) ≃ g ∘ K rel Y × ∂ I H \circ (g \times {1_I}) \simeq g \circ K\,{\text {rel}}\,Y \times \partial I .

Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An interpretation of these results in the context of Adams-Wilkerson embedding in integral K -theory is also given.

We give an enumeration of the set of homotopy classes of monomorphisms αm↪β2m-2 for vectorbundles α,β of dimensions m and (2m-2) over a differentiable closed manifold Mm of dimension m≥6. This is applied to the classification of immersions homotopic to a map g : Mm → N2m−2. In particular we can enumerate immersions from Mm in ℝ2m−2 for all m≥6.

This paper defines a group O(Mn, v0) which generalizes the group of framed homotopy n-spheres in Sn+k. Let Mn be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let v, be the normal bundle of some imbedding p: Mn -_ Sn + k, where 2k _ n + 3. Then O(Mn, v(,) is the set of h-cobordism classes of triples (F, Vn, f), where F: Sn + k T(v0) is a map which is transverse regular on M, Vn=F-1(Mn), and f=Ff Vn is a homotopy equivalence. (T(v,) is the Thom complex of v,.) There is a natural group structure on 0(Mn, v1,), and O(Mn, v1,) fits into an exact sequence similar to that for the framed homotopy n-spheres. This paper attempts to generalize in a natural way a well-known exact sequence concerning framed homotopy spheres which is contained in the work of Novikov [11], Kervaire-Milnor [7], and Levine [10]. The author stumbled onto these results partly because of his efforts to prove imbedding theorems for manifolds in the metastable range, and partly because of his recent work on Browder-Novikov theory for maps of degree d, Idl I=0 (see [2]). ?2 describes the basic constructions used in this paper. The "group of embedded manifolds", 6(M, v,), is defined in ?3. A fairly simple description of that group is given towards the end of that section. ?3 also contains the main results about O(M, v,). In ?4 we discuss a few interesting open problems. The author would like to thank the referee for some helpful suggestions. 1. Notation. All manifolds will be C O, compact, and oriented. Maps will be transverse to boundaries. If Mn is a connected closed manifold, let [M] E HnM denote the orientation class. Recall that f: Vn Mn is said to have degree d, i.e., degf=d, if f*([V]) =d[M], where f*: HnV HnM is the map induced byf on the integral homology groups. As usual, Dk denotes the closed unit ball in Euclidean k-space Rk, i.e., Dk={(yl, Yy) Rk|y2+I*...*+y2 O} and D_ ={(Y1, ,Yk+) eRk+l y2+ *?. +y2+ 1?=1,Y?<0}. We have natural inclusions S?kSk+l and DkCDk+l. Let e=(l 0) E SO?Sk. Received by the editors September 5, 1969. AMS 1970 subject classifications. Primary 57D55, 57D99; Secondary 57D60.

In this study, we introduce the concepts of fuzzy closure Hopf space and fuzzy closure Hopf group within the framework of fuzzy closure spaces, using homotopy theory. We investigate the relationships between the fuzzy closure Hopf group and its homotopy equivalence. Furthermore, we demonstrate the existence of a contravariant functor from the category of fuzzy closure Hopf spaces and the continuous functions, to the category of groups and homomorphisms. This is demonstrated by illustrating that the set of homotopy function classes among fuzzy closure Hopf groups constitutes a group.

In this paper, we study the Nisnevich sheafification \mathcal{H}^1_{\text{ét}}(G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G -torsors, for a reductive group G . We show that if G -torsors on affine lines are extended, then \mathcal{H}^1_{\text{ét}}(G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G -torsors. We also identify the sheaf \mathcal{H}^1_{\text{ét}}(G) with the sheaf of \mathbb{A}^1 -connected components of the classifying space \mathrm{B}_{\text{ét}}G . This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of \mathbb{A}^1 -connected components in terms of unramified G -torsors over function fields whenever Nisnevich-local purity holds for G -torsors.

Degree theory for continuous maps has a long history and has been extensively studied, both from the point of view of analysis and topology. If f ∈ C 0(S n, S n), deg f is a well-defined element of ℤ, which is stable under continuous deformation. Starting in the early 1980s, the need to define a degree for some classes of discontinuous maps emerged from the study of some nonlinear PDEs (related to problems in liquid crystals and superconductors). These examples involved Sobolev maps in the limiting case of the Sobolev embedding; see Sections 2 and 3 below. (Topological questions for Sobolev maps strictly below the limiting exponent have been investigated in [15] and [14].) In these cases, the Sobolev embedding asserts only that such maps belong to the space VMO (see below) and need not be continuous.

The multichannel rendezvous problem in cognitive radio networks (CRNs) has been a hot research topic lately. One of the most challenging settings of the multichannel rendezvous problem is the oblivious rendezvous problem in heterogeneous CRNs, where: 1) there are no distinguishable roles of users; 2) users’ clocks are not synchronized; 3) users may have different available channel sets; and 4) there is no universal labelling of the channels. Most existing works in the literature focus on achieving deterministic bounds for the maximum conditional time-to-rendezvous (MCTTR) and perform poorly (in comparison with the random algorithm) for the expected time-to-rendezvous (ETTR) due to the “stay” modes in these works. In this paper, we tackle the oblivious rendezvous problem by taking both MCTTR and ETTR into consideration. In order to have guaranteed rendezvous, we only make two assumptions: (A1) there is at least one common available channel and (A2) there is a unique ID for each user. We first propose a new class of strong symmetrization mappings to encode user IDs for speeding up the rendezvous process. Two efficient and yet simple encoding schemes are proposed by utilizing the $\cal C$ -transform and the existing 4B5B encoding. Based on the new class of strong symmetrization mappings, we propose the two-prime modular clock algorithm for the two-user rendezvous problem. The ETTR of our algorithm is almost the same as that of the random algorithm and its MCTTR is also comparable to the best existing bound. We also extend the two-prime modular clock algorithm for multiuser rendezvous by proposing the stick together algorithm and the spread out algorithm. One interesting finding for the multiuser rendezvous problem is that the spread out algorithm is not always better than the stick together algorithm as commonly claimed in the literature.

We provide a simple criterion for an element of the mapping class group of a closed surface to be a normal generator for the mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than \sqrt{2} is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.

Polynomial quadratic differentials on the complex plane and light-like polygons in the Einstein Universe

Recently, Frigon proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for this class of maps in a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by a modification of it, the interior condition. We also show that our arguments work for a certain class of generalized contractions, thus complementing a result of Agarwal and O'Regan.