Homotopy Groups Research Articles

External Spanier–Whitehead duality and homology representation theorems for diagram spaces

We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. We use this to construct Chern characters for certain rational $\mathcal{C}$-homology theories.

On the second homotopy group of the classifying space for commutativity in Lie groups

In this note we show that the second homotopy group of B(2, G), the classifying space for commutativity for a compact Lie group G, contains a direct summand isomorphic to $$\pi _1(G)\oplus \pi _1([G,G])$$ , where [G, G] is the commutator subgroup of G. It follows from a similar statement for E(2, G), the homotopy fiber of the canonical inclusion $$B(2,G)\hookrightarrow BG$$ . As a consequence of our main result we obtain that if E(2, G) is 2-connected, then [G, G] is simply-connected. This last result completes how the higher connectivity of E(2, G) resembles the higher connectivity of [G, G] for a compact Lie group G.

Codimension 2 transfer of higher index invariants

This paper is devoted to the study of the higher index theory of codimension 2 submanifolds originated by Gromov–Lawson and Hanke–Pape–Schick. The first main result is to construct the ‘codimension 2 transfer’ map from the Higson–Roe analytic surgery exact sequence of a manifold M to that of its codimension 2 submanifold N under some assumptions on homotopy groups. This map sends the primary and secondary higher index invariants of M to those of N. The second is to establish that the codimension 2 transfer map is adjoint to the co-transfer map in cyclic cohomology, defined by the cup product with a group cocycle. This relates the Connes–Moscovici higher index pairing and Lott’s higher $$\rho $$ -number of M with those of N.

A new family in the stable homotopy groups of spheres

In this paper, the nontriviality of homotopy elements β 1 ϱ s \beta _1\varrho _s in the p p -primary component of stable homotopy groups of spheres is shown by virtue of the Adams spectral sequence and the May spectral sequence, where ϱ s \varrho _s is represented by b 0 g 0 γ ~ s b_0g_{0}\widetilde {\gamma }_{s} in the Adams spectral sequence, p p is a prime number greater than 5 5 , and 3 ≤ s > p 3\leq s>p .

Stable homotopy groups

Stable homotopy groups

Growth of homotopy groups of spheres via the Goodwillie–EHP sequence

AbstractWe bound the volume of the homotopy groups of the 2‐local Goodwillie approximations of a sphere in terms of the amount of 2‐torsion in the stable stems, providing a Goodwillie‐theoretic refinement of a result of Burklund and Senger. At the ‐excisive approximation, this bound is obtained by ‘multiplying the stable answer by a polynomial of degree ’. The main tool is Behrens' Goodwillie–EHP long exact sequence.

Families of diffeomorphisms and concordances detected by trivalent graphs

AbstractWe study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. We specify some recent results and constructions of the second named author to show that those non‐trivial elements in homotopy groups are lifted to homotopy groups of the moduli space of ‐cobordisms . As a geometrical application, we show that those elements in for are also lifted to the rational homotopy groups of the moduli space of positive scalar curvature metrics. Moreover, we show that the same elements come from the homotopy groups of moduli space of concordances of positive scalar curvature metrics on with fixed‐round metric on the boundary .

On invertible $2$-dimensional framed and $r$-spin topological field theories

We classify invertible 2-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. By a recent result of Kreck, Stolz and Teichner the first homotopy groups are given by the so called SKK groups. We compute them explicitly using the combinatorial model of framed and $r$-spin surfaces of Novak, Runkel and the author.

Efficient simplicial replacement of semialgebraic sets

Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .

Algebraic Properties of 𝑷𝑮𝑳𝟐 (ℂ) for Long Exact Fibration Sequence with Sporadic Extensions

A concise formulation is given regarding the constructions of the group 𝑃𝐺𝐿2 (ℂ) with its related algebraic properties with intertwined topological aspects in the long exact fibration sequences as considered over homotopy and higher order homotopy groups with further extension to sporadic groups including the monster group formulations.

On homotopy groups of EChG24∧A1

Let $A_1$ be any spectrum in a class of finite spectra whose mod $2$ cohomology is isomorphic to a free module of rank one over the subalgebra $\mathcal{A}(1)$ of the Steenrod algebra. Let $E_{C}$ be the second Morava-$E$ theory associated to a universal deformation of the formal completion of the supersingular elliptic curve $(C) : y^{2}+y = x^{3}$ defined over $\mathbb{F}_{4}$ and $G_{24}$ a maximal finite subgroup of automorphism group $\mathbb{S}_{C}$ of the formal completion of $C$. In this paper, we compute the homotopy groups of $E_{C}^{hG_{24}}\wedge A_1$ by means of the homotopy fixed point spectral sequence.

On the space of initial values strictly satisfying the dominant energy condition

The dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce an index difference for initial value pairs and compare it to its classical counterpart for Riemannian metrics. Recent non-triviality results for the latter will then imply that this space has non-trivial homotopy groups.

Hodge theory for elliptic curves and the Hopf element ν$\nu$

AbstractWe show that the vector bundle on the moduli stack of elliptic curves associated to the 2‐cell complex is isomorphic to the de Rham cohomology sheaf of the universal elliptic curve . We use this to calculate the homotopy groups of the ‐quotient of by , called the spectrum of ‘topological quasimodular forms’, by relating its Adams–Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge–de Rham filtration.

Second homotopy groups of compact complex surfaces of non-general type

The aim of this paper is to compute the second homotopy groups of all compact, complex surfaces of non-general type.

Vietoris thickenings and complexes have isomorphic homotopy groups

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let \({\mathcal {U}}\) be a cover of a separable metric space X by open sets with a uniform diameter bound. The Vietoris complex \({\mathcal {V}}({\mathcal {U}})\) contains all simplices with vertex set contained in some \(U \in {\mathcal {U}}\), and the Vietoris metric thickening \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) is the space of probability measures with support in some \(U \in {\mathcal {U}}\), equipped with an optimal transport metric. We show that \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) and \({\mathcal {V}}({\mathcal {U}})\) have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover \({\mathcal {U}}\) appropriately, we get isomorphisms between the homotopy groups of Vietoris–Rips metric thickenings and simplicial complexes \(\pi _k(\textrm{VR}^\textrm{m}(X;r))\cong \pi _k(\textrm{VR}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using the convention “diameter \(< r\)” (instead of \(\le r\)). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes \(\pi _k(\check{\mathrm {{C}}}^\textrm{m}(X;r))\cong \pi _k(\check{\mathrm {{C}}}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using open balls (instead of closed balls).

On the product [formula omitted] in the stable homotopy groups of spheres

Let p be a prime greater than 5. In the present article, we show that the products α1β12β2γt are non-trivial in the p-primary components of the stable homotopy groups of spheres for 4<t<p.

Analytic Detection in Homotopy Groups of Smooth Manifolds

In this paper, for the mapping of a sphere into a compact orientable manifold Sn → M, n ≥ 1, we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold πn(M). For this purpose, we consistently use the theory of iterated integrals developed by 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 mappings f : S2n−1 → Sn, n ≥ 2.We give a brief description of Chen’s theory representing Whitehead’s and Haefliger’s 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.

Moduli spaces of compact RCD(0,N)-structures

The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of {{,mathrm{textrm{RCD}},}}(0,N)-structures. First, we relate the convergence of {{,mathrm{textrm{RCD}},}}(0,N)-structures on a space to the associated lifts’ equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of {{,mathrm{textrm{RCD}},}}(0,N)-structures that have non-trivial rational homotopy groups.

Dynamical mass generation for ferromagnetic skyrmions in two dimensions

Magnetic skyrmions are topological magnetization textures that are characterized by the homotopy group of two dimensional spheres. Despite years of intensive research on skyrmions, the fundamental problem of the inertia of a skyrmion in driven motion remains unresolved. By properly taking into account a direct coupling between skyrmion motion and the correspondingly excited magnons, we identify a dynamical mass for the skyrmion in motion. The direct coupling between skyrmion motion and magnons can be employed to engineer skyrmion dynamics with magnons through ingenious material and geometry design.

Tmf–based Mahowald invariants

The $2$-primary homotopy $\beta$-family, defined as the collection of Mahowald invariants of Mahowald invariants of $2^i$, $i \geq 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate $\mathit{tmf}$-based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of $\mathit{tmf}$ with trivial $C_2$-action and Behrens' filtered Mahowald invariant machinery.