J Group Research Articles

The relative James construction and its application to homotopy groups

In this paper, we develop the new method to compute the homotopy groups of the mapping cone Cf=Y∪fCX beyond the metastable range by analysing the homotopy of the n-th filtration of the relative James construction J(X,A) for CW-pair A↪iX, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map X∪iCA→ΣA. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod 2r Moore spaces for all positive integers r.

A short elementary proof of Beben and Theriault's theorem on homotopy fibers

Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. We give a short and elementary proof of this theorem.

Canonical Supermultiplets and Their Koszul Duals

The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document} algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type G2×A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2 \ imes A_1$$\\end{document}.

Digital h-Fibrations and Some New Results on Digital Fibrations

In this work, the notion of digital fiber homotopy is defined and its properties are given. We present some new results on digital fibrations. Moreover, we introduce digital h-fibrations. We prove some of the properties of these digital h-fibrations. We show that a digital fibration and a digital map p are fiber homotopic equivalent if and only if p is a digital h-fibration. Finally, we explore a relation between digital fibrations and digital h-fibrations.

H-Principle for Stratified Spaces

Abstract We extend Gromov and Eliashberg–Mishachev’s $h-$principle on manifolds to stratified spaces. This is done in both the sheaf-theoretic framework of Gromov and the smooth jets framework of Eliashberg–Mishachev. The generalization involves developing 1. the notion of stratified continuous sheaves to extend Gromov’s theory, 2. the notion of smooth stratified bundles to extend Eliashberg–Mishachev’s theory. A new feature is the role played by homotopy fiber sheaves. We show, in particular, that stratumwise flexibility of stratified continuous sheaves along with flexibility of homotopy fiber sheaves furnishes the parametric $h$-principle. We extend the Eliashberg–Mishachev holonomic approximation theorem to stratified spaces. We also prove a stratified analog of the Smale–Hirsch immersion theorem.

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.

Amenability and acyclicity in bounded cohomology

Johnson’s characterization of amenable groups states that a discrete group \Gamma is amenable if and only if H_b^{n \geq 1}(\Gamma; V) = 0 for all dual normed \mathbb{R}[\Gamma] -modules V . In this paper, we extend the previous result to homomorphisms by proving the converse of the mapping theorem : a surjective group homomorphism \phi \colon \Gamma {\to} K has amenable kernel H if and only if the induced inflation map H^\bullet_b(K; V^H) {\to} H^\bullet_b(\Gamma; V) is an isometric isomorphism for every dual normed \mathbb{R}[\Gamma] -module V . In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms \phi \colon \Gamma \to K with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all Banach \Gamma -modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms, that is, the class of group homomorphisms \phi \colon \Gamma \to K for which the restriction maps in bounded cohomology H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \phi^{-1}V) are isomorphisms for a suitable family of dual normed \mathbb{R}[K] -modules V including the trivial \mathbb{R}[K] -module \mathbb{R} . We then extend the first and third results to topological spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.

The positive scalar curvature cobordism category

We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside R+(Sd) if d≥6. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fiber of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen’s Theorem B and instances of the Gromov–Lawson surgery theorem. We extend some of the surgery arguments by Galatius and the second-named author to the psc setting to pass between different connectivity conditions. Segal’s theory of Γ-spaces is then used to construct the claimed infinite loop space structures. The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen–Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontryagin classes. A delooped version of the Atiyah–Singer index theorem proved by the first-named author is, moreover, used to show that the secondary index invariant to real K-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.

The stable Adams conjecture and higher associative structures on Moore spectra

In this paper, we provide a new proof of the stable Adams conjecture. Our proof constructs a canonical null-homotopy of the stable J-homomorphism composed with a virtual Adams operation, by applying the K-theory functor to a multinatural transformation. We also point out that the original proof of the stable Adams conjecture is incorrect and present a correction. This correction is crucial to our main application. We settle the question on the height of higher associative structures on the mod $p^k$ Moore spectrum $\mathrm{M}_p(k)$ at odd primes. More precisely, for any odd prime $p$, we show that $\mathrm{M}_p(k)$ admits a Thomified $\mathbb{A}_n$-structure if and only if $n \lt p^k$. We also prove a weaker result for $p=2$.

A new approach to model categorical homotopy fiber sequences

We propose a simplified definition of Quillen’s fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding that the homotopy

Koszul complexes over Cohen-Macaulay rings

We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and a1,…,an is any sequence of elements in A, then the Koszul complex K(A;a1,…,an) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring A, by finding a Cohen-Macaulay DG-ring B such that H0(B)=A, and using the Cohen-Macaulay structure of B to deduce results about A. As application, we prove that if f:X→Y is a morphism of schemes, where X is Cohen-Macaulay and Y is nonsingular, then the homotopy fiber of f at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.

Topological and geometrical aspects of band theory

This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. Amusingly, many of these developments have a connection to contributions in high-energy physics by Dirac. The review starts by a presentation of the Dirac magnetic monopole, goes on with the Berry phase in a two-level system and the geometrical/topological band theory for Bloch electrons in crystals. Next, specific examples of tight-binding models giving rise to lattice versions of the Dirac equation in various space dimension are presented: in 1D (Su–Schrieffer–Heeger (SSH) and Rice–Mele models), 2D (graphene, boron nitride, Haldane model) and 3D (Weyl semi-metals). The focus is on topological insulators and topological semi-metals. The latter have a Fermi surface that is characterized as a topological defect. For topological insulators, the two alternative view points of twisted fiber bundles and of topological textures are developed. The minimal mathematical background in topology (essentially on homotopy groups and fiber bundles) is provided when needed. Topics rarely reviewed include: periodic versus canonical Bloch Hamiltonian (basis I/II issue), Zak versus Berry phase, the vanishing electric polarization of the SSH model and Dirac insulators.

The fibre of the degree 3 map, Anick spaces and the double suspension

AbstractLet S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

Addition formulae for Nielsen numbers and Nielsen type numbers of fibre preserving maps defined locally

In 1995 Heath, Keppelmann and Wong studied addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps. In this paper we generalize these ideas to the setting of fibre preserving maps defined locally. Let (E,p,B) be a fibre space with E,B and all fibres compact connected ANR's, V be an open subset in B, and U=p−1(V). Let f:U→E be a fibre preserving map, i.e., a map for which there exists a map f¯:V→B with pf=f¯pU,V, where pU,V:U→V is the restriction of p. We give necessary and sufficient conditions for when a naïve addition formula expressing the local Nielsen number n(f;U) as a simple sum of Nielsen numbers on the fibres is valid. We also introduce a Nielsen type number nF(f;U,p) for a fibre preserving map f:U→E, we show that it is a lower bound for the least number of fixed points within the fibre homotopy class of f. Some calculations and examples are given.

Whitehead products in moment-angle complexes

In toric topology, to a simplicial complex $K$ with $m$ vertices, one associates two spaces, the moment-angle complex $\mathcal{Z}_K$ and the Davis–Januszkiewicz space $DJ_K$. These spaces are connected by a homotopy fibration $\mathcal{Z}_K \to DJ_K \to (\mathbb{C} P^{\infty})^m$. In this paper, we show that the map $\mathcal{Z}_K \to DJ_K$ is identified with a wedge of iterated (higher) Whitehead products for a certain class of simplicial complexes $K$ including dual shellable complexes. We will prove the result in a more general setting of polyhedral products.

K-theoretic Tate–Poitou duality and the fiber of the cyclotomic trace

Let $$p\in {\mathbb {Z}}$$ be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace $$K({\mathcal {O}}_{F})^{\scriptscriptstyle \wedge }_{p}\mathchoice{\longrightarrow }{\rightarrow }{\rightarrow }{\rightarrow }TC({\mathcal {O}}_{F})^{\scriptscriptstyle \wedge }_{p}$$ after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of $${\mathbb {Z}}$$ .

Topological $K$-theory with coefficients and the $e$-invariant

We compare the invariants of flat vector bundles defined by Atiyah et al. and Jones et al. and prove that, up to weak homotopy, they induce the same map from the 0-connective algebraic K-theory space of the complex numbers to the homotopy fiber of the Chern character. We examine homotopy properties of this map and its relation with other known invariants. In addition, using the formula for ξ̃-invariants of lens spaces derived from Donnelly’s fixed point theorem and the 4-dimensional cobordisms constructed via relative Kirby diagrams, we recover the formula of the e-invariants of Seifert homology spheres given by Jones and Westbury, up to sign.

Classifying spaces for commutativity of low-dimensional Lie groups

AbstractFor each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

The Homotopy Types of Sp(2)-Gauge Groups over Closed Simply Connected Four-Manifolds

We determine the number of distinct fibre homotopy types for the gauge groups of principal $Sp(2)$-bundles over a closed, simply-connected four-manifold.

A note on 3-manifolds and complex surface singularities

This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical $$\mathrm{SU}(2)$$-frame. To start we notice that the set of homotopy classes of $$\mathrm{SU}(2)$$-frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a $$\mathbb Z$$-torsor. Then we define the $$\widehat{E}$$-invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The $$\widehat{E}$$-invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities.