Relative Homotopy Groups Research Articles

Relative homotopy groups and Serre fibrations for polynomial maps

Relative homotopy groups and Serre fibrations for polynomial maps

Quantum anomalous semimetals

The topological states of matter and topological materials have been attracting extensive interests as one of the frontier topics in condensed matter physics and materials science since the discovery of quantum Hall effect in 1980s. So far all the topological phases such as integer quantum Hall effect and topological insulators are characterized by integer topological invariants. None is a half integer or fractional. Here we propose a type of semimetals which hosts a single cone of Wilson fermions. The Wilson fermions possess linear dispersion near the Dirac point, but break the chiral or parity symmetry such that an unpaired Dirac cone can be realized on a lattice. In order to avoid the fermion doubling problem, the chiral symmetry or parity symmetry must be broken explicitly if the hermiticity, locality and translational invariance all hold. We find that the system can be classified by the relative homotopy group, and a half-integer topological invariant. We term the nontrivial quantum phase as quantum anomalous semimetal. The work opens the door towards exploring novel states of matter with fractional topological charge.

One-dimensional nexus objects, network of Kibble-Lazarides-Shafi string walls, and their spin dynamic response in polar-distortedB-phase ofHe3

The Kibble-Lazarides-Shafi (KLS) domain wall problem in the axion solution of CP violation in QCD has condensed-matter based analogy in the nafen-distorted superfluid $^3$He. Recent experiment in rotating superfluid $^3$He produced the network of KLS string walls in human controllable system. In this system, KLS string wall appears in two-step symmetry break transition from normal phase to polar-distorted B phase and turn out to be the descendants of HQVs of polar phase. Here we show the KLS string wall smoothly connects to spin solitons when the spin orbital coupling is taken into account. This means HQVs are 1D nexus which connects the spin solitons and the KLS domain walls. This is because the subgroup $G=\pi_{1}(S_{S}^{1},\tilde{R}_{2})$ of relative homotopy group describing the spin solitons is isomorphic to the group describing the half spin vortices -- the spin textures KLS string wall. In the nafen-distorted $^3$He system, 1D nexus objects and the spin solitons with topological invariant $2/4$ form two different types of network, which are named as pseudo-random lattices of inseparable and separable spin solitons. These two types lattices correspond to two different representations of $G$. We discuss the condition under which pseudo-random lattices model works. The equilibrium configuration and surface densities of free energies are calculated by numeric minimization. Based on the equilibrium spin textures of different pseudo-random lattices, we calculated their transverse NMR spectrum, the resulted frequency shifts and $\sqrt{\Omega}$-scaling of ratio intensity exactly coincide with the experimental measurements. We also discussed the mirror symmetry in the presence of KLS domain wall and the influence of the explicitly break of this discrete symmetry. Our discussions and considerations can be applied to the composite defects in other condensed matter and cosmological system.

String monopoles, string walls, vortex skyrmions, and nexus objects in the polar distorted B phase of He3

The composite cosmological objects -- Kibble-Lazarides-Shafi (KLS) walls bounded by strings and cosmic strings terminated by Nambu monopoles -- could be produced during the phase transitions in the early Universe. Recent experiments in superfluid $^3$He reproduced the formation of the KLS domain walls, which opened the new arena for the detailed study of those objects in human controlled system with different characteristic lengths. These composite defects are formed by two successive symmetry breaking phase transitions. In the first transition the strings are formed, then in the second transition the string becomes the termination line of the KLS wall. In the same manner, in the first transition monopoles are formed, and then in the second transition these monopoles become the termination points of strings. Here we show that in the vicinity of the second transition the composite defects can be described by relative homotopy groups. This is because there are two well separated length scales involved, which give rise to two different classes of the degenerate vacuum states, $R_1$ and $R_2$, and the composite objects correspond to the nontrivial elements of the group $\pi_n(R_1,R_2)$. We discuss this on example of the so-called polar distorted B phase, which is formed in the two-step phase transition in liquid $^3$He distorted by aerogel. In this system the string monopoles terminate spin vortices with even winding number, while KLS string walls terminate on half quantum vortices. In the presence of magnetic field, vortex-skyrmions are formed, and the string monopole transforms to the nexus. We also discuss the integer-valued topological invariants of those objects. Our consideration can be applied to the composite defects in other condensed matter and cosmological systems.

Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-2 and four period-8 cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme, the classification in arbitrary dimensions $d\ensuremath{\ge}1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys. Rev. B 90, 165114 (2014)]. The use of a scattering-matrix-inspired method allows us to address the second descendant ${\mathbb{Z}}_{2}$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.

Asphericity results for ribbon disk complements via alternate descriptions

An alternate description for ribbon disk complements in the $4$-ball is provided. It is known (and reestablished) that this description is equivalent to the standard LOT description, up to $3$-deformation. Amenable to geometric arguments, the alternate description yields asphericity results for ribbon disk complements using simple graph-theoretic criteria and, later, using a relative homotopy group which arises naturally. In the course of making and modifying the description, two algorithms are given for presenting ribbon disk groups.

On Relative Homotopy Groups of Modules

In his book “Homotopy Theory and Duality,” Peter Hilton described the concepts of relative homotopy theory in module theory. We study in this paper the possibility of parallel concepts of fibration and cofibration in module theory, analogous to the existing theorems in algebraic topology. First, we discover that one can study relative homotopy groups, of modules, from a viewpoint which is closer to that of (absolute) homotopy groups. Then, through the study of various cases, we learn that the classic fibration/cofibration relation does not come automatically. Nonetheless, the ability to see the relative homotopy groups as absolute homotopy groups, in a stronger sense, promises to justify our ultimate search.

Relations between defects in the bulk and on the surface of an ordered medium: a topological investigation

Starting from a proposal of Volovik (1978) we investigate surface point, line, and wall defects with the aid of relative homotopy groups. The exact sequence of homotopy groups is used in interpreting the relations between surface and bulk singularities. In particular, we consider the possibilities for surface singularities to move into the bulk, for bulk singularities to leave the medium through the surface, and for singular loops to be broken apart by the surface. The theory is extended to the case where the surface induces a thermodynamic phase differing from that in the bulk. An example is given of a biaxial nematic surface with the Klein bottle as the order-parameter space. Due to the fundamental theorem for homotopy groups of fibre bundles, the classification scheme of surface defects is identical for all pairs of bulk and surface order-parameter spaces which are related by an inverse-bundle projection.

On relative homotogy groups of the product filtration, the James construction, and a formula of Hopf

We compute the nth relative homotopy group of ( Z n , Z n−1 ) when Z ∗ is a product of certain filtered spaces. A consequence is information on the homotopy of ΩΣX when X is the classifying space of a crossed complex.

Lectures on Homotopy Theory

1. Homotopy Groups. 1.1 Function Spaces. 1.2 H-Spaces and CoH-Spaces. 1.3 Homotopy Groups. 2. Fibrations and Cofibrations. 2.1 Pullbacks and Pushouts. 2.2 Fibrations. 2.3 Cofibrations. 2.4 Applications of the Mapping Cylinder. 3. Exact Homotopy Sequences. 3.1 Exact Sequence of a Map: Covariant Case. 3.2 Exact Sequence of a Map: Contravariant Case. 4. Simplicial Complexes. 4.1 Simplicial Complexes. 4.2 Simplicial Approximation Theorem. 4.3 Polyhedra. 4.4 Fibrations and Polyhedra. 5. Relative Homotopy Groups. 5.1 Homotopy Groups of Maps. 5.2 Quasifibrations. 5.3 Some Homotopy Groups of spheres. 6. Homotopy Theory of CW-complexes. 6.1 CW-complexes. 6.2 Homotopy theory of CW-complexes. 6.3 Eilenberg-Mac Lane spaces. 7. Fibrations revisited. 7.1 Sections of fibrations. 7.2 F -fibrations. 7.3 Universal F -fibrations. Appendix: Colimits. Compactly generated spaces. Index.

Van Kampen theorems for diagrams of spaces

where @ means “non-Abelian tensor product” of the two relative homotopy groups, each acting on the other via ;r,C. This new algebraic construction M @ N, which is defined for a pair of groups M, N each of which acts on the other, is studied in $2. As is well-known, a general determination of a triad homotopy group has consequences for certain absolute homotopy groups. Some of these are given in $3. For example, we prove that for any group G

Free simplicial groups and the second relative homotopy group of an adjunction space

(If (YX) is a pair of spaces, then 7rnz(YX) is a ;rrtX-crossed module.) Suppose that X is a connected CW-complex, and that Y is obtained from X by attaching 2-cells. Whitehead [9] showed that in this case, x2(xX) is a free x,X-crossed module, i.e., if {c,} are the elements of ;71#‘,X) corresponding to the 2-cells of Y-X, and if (g,} are elements of another n,X-crossed module G with ag,=ac,, then there is a unique homomorphism of crossed modules h : 7t2(Y,X) + G for which h(c,) =g,. Whitehead’s proof of this theorem is rather difficult and geometric (see [l] for a more modern exposition of Whitehead’s proof). Brown and Higgins [2] have shown that this theorem follows from their 2-dimensional generalization of the van Kampen theorem, in the context of ‘double groupoids’. Ratcliffe [7], using his study of free and projective crossed modules, was able to give a more algebraic proof. The purpose of the present paper is to give a completely algebraic proof of this theorem. We will obtain the theorem through a study offree simplicial groups, which are models for loop spaces (and are therefore useful for computing (absolute) homotopy groups).

Colimit theorems for relative homotopy groups

The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω-groupoids and C of crossed complexes were defined, and the principal result of [10] was an equivalence of categories γ : G → C. Also established were a version of the homotopy addition lemma, and properties of ‘thin’ elements, in an ω-groupoid. In particular it was proved that an ω-groupoid is a special kind of Kan cubical complex, in that every box has a unique thin filler. All these results will be used here.

On the Second Relative Homotopy Group of an Adjunction Space: An Exposition of a Theorem of J. H. C. Whitehead

In his 1949 paper [6], a different exposition of part of the proof was given, and also the result was codified finally in saying that the group A is the free crossed nx{XQ, x0)-module on the 2-cells. One difficulty in obtaining this theorem by standard methods of algebraic topology is that it is a non-abelian result. A proof has recently been given by Ratcliffe, using a homological characterisation of free crossed modules [3]. The theorem also is a special case of the generalisation to dimension 2 of the Seifert-van Kampen Theorem [1], where free crossed modules arise as very special cases of pushouts of crossed modules. Indeed one aim of the program completed in [1] was to forpulate a generalisation of Whitehead's theorem, and to prove it by verification of a universal property. A list of papers which apply the theorem is also given in [1]. In spite of these other proofs and generalisations, Whitehead's proof still has interest. It has a curious structure, a modern-looking use of transversality, and a clever interplay of the relations for a knot group and the rules of a crossed module. Also the /ideas have some relevance to difficult problems in combinatorial group theory involving identities between relations. (I hope to give elsewhere an exposition of some ideas of Peter Stefan in this area.t) But the proof is difficult to read, for reasons which include its originality of conception, and the change in notation and formulation over the years 1941-49. I hope therefore that it will prove useful to present a straightforward account of Whitehead's proof; my own contribution is simply that of presentation in a modern, and uniform notation and terminology. Thanks are due to the late Peter Stefan for discussions on some of this material, and to Johannes Huebschmann for helpful comments on an earlier draft.

Fitted diffeomorphisms of non-simply connected manifolds

TOPOLOGICAL ENTROPY roughly speaking measures the complexity of a dynamical system. In “Homology Theory and Dynamical Systems”[ l] Shub and Sullivan defined a class of structurally stable diffeomorphisms called fitted which are Co dense in oil?‘(M). They show how to relate the dynamics of a fitted diffeomorphism to the induced chain map in homology theory, and using this determine the minimum topological entropy of fitted diffeomorphisms in each component of D%‘(M), for simply connected manifolds of dimension greater than five. In this paper we extend these results to the non simply-connected case. The main tools are a sharpened statement of the Whitney canceIling lemma for handles of index 2 and (n 2) using relative homotopy groups, and classical results of Whitelead concerning these groups[2]. Our methods also lead to an algebraic characterization of the chain complexes over Z[II,] which arise from handle decompositions of high dimensional manifolds, extending a theorem of Smale’s in the simply connected case. It is a pleasure to acknowledge many helpful conversations with Michael Shub and Dennis Sullivan.

Topology of non-singular textures of superfluid3He

The topological classification of non-singular textures of superfluid 3He is discussed using relative homotopy groups.

On the Connection between the Second Relative Homotopy Groups of Some Related Spaces

Proceedings of the London Mathematical SocietyVolume s3-36, Issue 2 p. 193-212 Articles On the Connection between the Second Relative Homotopy Groups of Some Related Spaces Ronald Brown, Ronald Brown School of Mathematics and Computer Science, University College of North Wales, Bangor, LL57 2UW GwyneddSearch for more papers by this authorPhilip J. Higgins, Philip J. Higgins Department of Mathematics, King's College, Strand, London, WC2R 2LSSearch for more papers by this author Ronald Brown, Ronald Brown School of Mathematics and Computer Science, University College of North Wales, Bangor, LL57 2UW GwyneddSearch for more papers by this authorPhilip J. Higgins, Philip J. Higgins Department of Mathematics, King's College, Strand, London, WC2R 2LSSearch for more papers by this author First published: March 1978 https://doi.org/10.1112/plms/s3-36.2.193Citations: 45AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes3-36, Issue2March 1978Pages 193-212 RelatedInformation

Embeddings of bounded manifolds

Introduction. This paper gives some embedding and unknotting theorems for bounded manifolds in the PL and differential categories. The theorems are of a similar nature to the embedding and unknotting theorems of Irwin and Zeeman (9, 18), except that the boundaries are allowed to move and the appropriate connectivity conditions become conditions on the relative homotopy groups of the manifolds modulo the boundary, or some suitable part of the boundary.

The Homotopy Groups of a Triad

induce isomorphisms of the relative homology groups in all dimensions. Simple examples show that this need not be true for the relative homotopy groups, even when X is a finite connected simplicial complex, and A, B, and A n B are connected subcomplexes. The new homotopy groups defined in this paper are a measure of the amount by which the excision axiom fails to hold for relative homotopy groups. The precise meaning of this statement will be clear later. One special case for which it is particularly important to determine the extent of the validity of the excision axiom for relative homotopy groups is the following: Let K be a cell complex2 and let K', n 0, 1, 2, * * , denote the n-skeleton. Denote the closed n-cells of K by al', a, .* , their boundaries by al , &n, **, and set

On Operators in Relative Homotopy Groups

be a deformation in which ft(ao) may wander, provided fi(ao) = fo(ao) -po, and let ar e 7r, (P) be the element represented by the map 0: (I, 0, 1) -+ (P, Po, Po) (I = I), where 0(1 t) = ft(ao). Let2 ar be the element of 7n(Q, P), which is represented by the map fi. The transformation a -* ra, for a fixed ar e 7r,(P) and variable a e 7rw(Q, P), is an automorphism, Ta, of 7rn(Q, P). Also 0f + Tis a homomorphism of 7r,(P) in the group of automorphisms of 7rn(Q, P). Thus 7rn(Q, P) is a group with operators, the operators being in 7rj(P). If P = po, so that 7r,(Q, P) is the absolute homotopy group, rn(Q), the above process only produces the identity. In this case an automorphism a -* aa, with a e Xrn.(Q)a ? r1i(P), is defined' by means of a deformation ft: In -* Q such that ft(In) 0(1 t), where 0: (1, 0, 1) --+Q, po, po) represents ae 7ri(Q). Thus we have two distinct processes for defining operators. The purpose of this note is to give a generalization of the second process from absolute to relative homotopy groups. The method is to deform the identity map i: P -+ P by means of a homotopy it: P -* Q, such that4 {l = i and 4o is a map of the form 4o: (P, po) -* (P, po). Let fo be given by (1.1), let go = fo In and let gt = 6_,go . Then the homotopy gt may be extended throughout In to give a deformation ft: In -* Q and the analogue of 0a e 7rn(Q) is the element of 7rn(Q, P), which is represented by the map fi .