Effectual topological complexity

We introduce the effectual topological complexity (ETC) of a [Formula: see text]-space [Formula: see text]. This is a [Formula: see text]-equivariant homotopy invariant sitting in between the effective topological complexity of the pair [Formula: see text] and the (regular) topological complexity of the orbit space [Formula: see text]. We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the nontrivial obstruction responsible for the fact that the topological complexity of the Klein bottle is [Formula: see text]. In addition, this gives a counterexample to the possibility — suggested in Pavešić’s work on the topological complexity of a map — that ETC of [Formula: see text] would agree with Farber’s [Formula: see text] whenever the projection map [Formula: see text] is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners.

Analytic Detection in Homotopy Groups of Smooth Manifolds

In this paper, for the mapping of a sphere into a compact orientable manifoldSnM,n1, we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifoldn(M) n(M ). For this purpose, we consistently use the theory of iterated integrals developed by K.-T. Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappingsf:S2n-1Sn,n2. We give a brief description of Chens theory, representing Whiteheads and Haefligers formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group.

Dirac/Rarita–Schwinger plus Maxwell theories in ℝ × S3 spacetime in the Hopf coordinates

We consider the sets of Dirac–Maxwell and Rarita–Schwinger–Maxwell equations in [Formula: see text] spacetime. Using the Hopf coordinates, we show that these equations allow separation of variables and obtain the corresponding analytic and numerical solutions. It is also demonstrated that the current of the Dirac field is related to the Hopf invariant on the [Formula: see text] fibration.

Spinor field solutions in F(B2) modified Weyl gravity

We consider modified Weyl gravity where a Dirac spinor field is nonminimally coupled to gravity. It is assumed that such modified gravity is some approximation for the description of quantum gravitational effects related to the gravitating spinor field. It is shown that such a theory contains solutions for a class of metrics which are conformally equivalent to the Hopf metric on the Hopf fibration. For this case, we obtain a full discrete spectrum of the solutions and show that they can be related to the Hopf invariant on the Hopf fibration. The expression for the spin operator in the Hopf coordinates is obtained. It is demonstrated that this class of conformally equivalent metrics contains the following: (a) a metric describing a toroidal wormhole without exotic matter; (b) a cosmological solution with a bounce and inflation and (c) a transition with a change in metric signature. A physical discussion of the results is given.

Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

After recent work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem [M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. Math. (2), 184 (2016), 1–262], as well as Adams’ solution of the Hopf invariant one problem [J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (2), 72 (1960), 20–104], an immediate consequence of Curtis conjecture is that the set of spherical classes in H∗Q0S0 is finite. Similarly, Eccles conjecture, when specialized to X=Sn with n> 0, together with Adams’ Hopf invariant one theorem, implies that the set of spherical classes in H∗QSn is finite. We prove a filtered version of the above finiteness properties. We show that if X is an arbitrary CW-complex of finite type such that for some n, HiX≃0 for any i>n, then the image of the composition π∗ΩlΣl+2X→π∗QΣ2X→H∗QΣ2X is finite; the finiteness remains valid if we formally replace X with S−1. As an application, we provide a lower bound on the dimension of the sphere of origin on the potential classes of π∗QSn which are detected by homology. We derive a filtered finiteness property for the image of certain transfer maps ΣdimgBG+→QS0 in homology. As an application to bordism theory, we show that for any codimension k framed immersion f:M↬ℝn+k which extends to an embedding M→ℝd×ℝn+k, if n is very large with respect to d and k then the manifold M as well as its self-intersection manifolds are boundaries. Some results of this paper extend results of Hadi [Spherical classes in some finite loop spaces of spheres. Topol. Appl., 224 (2017), 1–18] and offer corrections to some minor computational mistakes, hence providing corrected upper bounds on the dimension of spherical classes H∗ΩlSn+l. All of our results are obtained at the prime p = 2.

Hopfions

More than 40 years ago, Faddeev proposed the existence of three-dimensional topological solitons classified by the integer-valued Hopf invariant. These solitons are now known as hopfions and have been investigated in a range of systems, including the original model suggested by Faddeev, where a variety of stable knot and link solutions have been computed numerically. Very recently, numerical computations have predicted the existence of nanoscale hopfions in frustrated magnets and experiments have realized micrometer-sized hopfions in chiral ferromagnetic fluids. All these examples of hopfions will be described and their similarities and differences discussed.