Homotopy Theory Research Articles

Ambidexterity in chromatic homotopy theory

We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the infty -categories of T!left( nright) -local spectra are infty -semiadditive for all n, where T!left( nright) is the telescope on a v_{n}-self map of a type n spectrum. This generalizes and provides a new proof for the analogous result of Hopkins–Lurie on K!left( nright) -local spectra. Moreover, we show that K!left( nright) -local and T!left( nright) -local spectra are respectively, the minimal and maximal 1-semiadditive localizations of spectra with respect to a homotopy ring, and that all such localizations are in fact infty -semiadditive. As a consequence, we deduce that several different notions of “bounded chromatic height” for homotopy rings are equivalent, and in particular, that T!left( nright) -homology of pi -finite spaces depends only on the nth Postnikov truncation. A key ingredient in the proof of the main result is a construction of a certain power operation for commutative ring objects in stable 1-semiadditive infty -categories. This is closely related to some known constructions for Morava E-theory and is of independent interest. Using this power operation we also give a new proof, and a generalization, of a nilpotence conjecture of J. P. May, which was proved by A. Mathew, N. Naumann, and J. Noel.

Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories

This paper provides a detailed study of 4-dimensional Chern-Simons theory on mathbb {R}^2times mathbb {C}P^1 for an arbitrary meromorphic 1-form omega on mathbb {C}P^1. Using techniques from homotopy theory, the behaviour under finite gauge transformations of a suitably regularised version of the action proposed by Costello and Yamazaki is investigated. Its gauge invariance is related to boundary conditions on the surface defects located at the poles of omega that are determined by isotropic Lie subalgebras of a certain defect Lie algebra. The groupoid of fields satisfying such a boundary condition is proved to be equivalent to a groupoid that implements the boundary condition through a homotopy pullback, leading to the appearance of edge modes. The latter perspective is used to clarify how integrable field theories arise from 4-dimensional Chern-Simons theory.

A Model of Directed Graph Cofiber

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective.

Inertia groups and smooth structures on quaternionic projective spaces

Abstract This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

A new kind of $F$-contraction and some best proximity point results for such mappings with an application

In this paper, we aim to present a new and unified way, including the previously mentioned solution methods, to overcome the problem in [7] for closed and bounded valued $F$-contraction mappings. We also want to obtain a real generalization of fixed point results existing in the literature by using best proximity point theory. Further, considering the strong relationship between homotopy theory and various branches of mathematics such as category theory, topological spaces, and Hamiltonian manifolds in quantum mechanics, our objective is to present an application to homotopy theory of our best proximity point results obtained in the paper. In this sense, we first introduce a new family, which is larger than $\mathcal{F}^\ast$ that has often been used to give a positive answer to the problem. Then, we prove some best proximity point results for the new kind of $F$% -contractions on quasi metric spaces via the new family. Additionally, we show that the note given by Almeida et al. [4] is not valid for our results. Therefore, our results are real generalizations of fixed point results in the literature. Moreover, we give comparative examples to demonstrate that our results unify and generalize some well-known results in the literature. As an application, we show that each homotopic mapping to $\varphi$ satisfying all the hypotheses of our best proximity point result has also a best proximity point.

Immersions of manifolds and homotopy theory

Immersions of manifolds and homotopy theory

On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds

Abstract We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.

Global stable splittings of Stiefel manifolds

We prove global equivariant refinements of Miller's stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces $O/O(m)$, $U/U(m)$ and $Sp/Sp(m)$. As such, our results encode compatible equivariant stable splittings, for all compact Lie groups, of specific equivariant refinements of these spaces. In the unitary and symplectic case, we also take the actions of the Galois groups into account. To properly formulate these Galois-global statements, we introduce a generalization of global stable homotopy theory in the presence of an extrinsic action of an additional topological group.

Hyperoctahedral homology for involutive algebras

Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group.

Representation stability and outer automorphism groups

In this paper we study families of representations of the outer automorphism groups indexed on a collection of finite groups \mathcal{U} . We encode this large amount of data into a convenient abelian category which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of T. Church et al. [Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)], we investigate for which choices of \mathcal{U} the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibit representation stability.

Some fixed point results via auxiliary functions on orthogonal metric spaces and application to homotopy

<abstract><p>In 2017, the concepts of orthogonal set and orthogonal metric spaces are presented. And an extension of Banach fixed point theorem is proved in this type metric spaces. Further in 2019, on orthogonal metric spaces, some fixed point theorems via altering distance functions are investigated. In this paper, presence and uniqueness of fixed points of the generalizations of contraction principle via auxiliary functions are investigated. And some consequences and an illustrative example are presented. On the other hand, homotopy theory constitute an important area of algebraic topology, but the application of fixed point results in orthogonal metric spaces to homotopy has not been done until now. As a different application in this field, the homotopy application of the one of the corollaries is given at the end of this paper.</p></abstract>

Realisability of the group of self-homotopy equivalences and local homotopy theory

Realisability of the group of self-homotopy equivalences and local homotopy theory

N∞-operads and associahedra

We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the collection of N-infinity-operads stands in bijection with the poset structure of the (n+1)-associahedron. We further provide a lower bound for the number of possible N-infinity-operads for any finite cyclic group G.

Combinatorial N∞ operads

We prove that the homotopy theory of $N_\infty$ operads is equivalent to a homotopy theory of discrete operads, and we construct free and associative operadic realizations of every indexing system. This resolves a conjecture of Blumberg and Hill in the affirmative.

English

A new approach to etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\mathcal{X}$ is an arbitrary higher stack on the etale site of affine schemes of finite type over $\mathbb{C},$ then the etale homotopy type of $\mathcal{X}$ agrees with the homotopy type of the underlying stack $\mathcal{X}_{top}$ on the topological site, after profinite completion. In particular, if $\mathcal{X}$ is an Artin stack locally of finite type over $\mathbb{C}$, our definition of the etale homotopy type of $\mathcal{X}$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\mathcal{X}_{top}$ of $\mathcal{X}$ in the sense of Noohi. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest.

English

We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real representation of a group $\mathsf{G}$, or more generally a finite categorical group, on a linear category, we associate a number, the modified secondary trace, to each graded commuting pair $(g, \omega) \in \mathsf{G} \times \hat{\mathsf{G}}$, where $\hat{\mathsf{G}}$ is the background Real structure on $\mathsf{G}$. This collection of numbers defines the Real $2$-character of the Real representation. We also define various forms of induction for Real representations of finite categorical groups and compute the result at the level of Real $2$-characters. We interpret results in Real categorical character theory in terms of geometric structures, namely gerbes, vector bundles and functions on iterated unoriented loop groupoids. This perspective naturally leads to connections with the representation theory of unoriented versions of the twisted Drinfeld double of $\mathsf{G}$ and with discrete torsion in $M$-theory with orientifold. We speculate on an interpretation of our results as a generalized Hopkins--Kuhn--Ravenel-type character theory in Real equivariant homotopy theory.

The homotopy theory of complete modules

Given a commutative ring R and finitely generated ideal I, one can consider the classes of I-adically complete, L0I-complete and derived I-complete complexes. Under a mild assumption on the ideal I called weak pro-regularity, these three notions of completions interact well. We consider the classes of I-adically complete, L0I-complete and derived I-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism R→S, we then give necessary and sufficient conditions for the categories of complete R-complexes and the categories of complete S-complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology.

Homotopy theory of Moore flows (II)

This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.

Solving binary programming problems using homotopy theory ideas

PurposeThe aim of the study is to show that merging two areas of mathematics – topology and discrete optimization – could result in a viable option to solve classical or specialized integer problems.Design/methodology/approachIn the paper, discrete topology concepts are applied to propose a metaheuristic algorithm that is capable to solve binary programming problems. Particularly, some of the homotopy for paths principles are used to explore the solution space associated with four well-known NP-hard problems herein considered as follows: knapsack, set covering, bi-level single plant location with order and one-max.FindingsComputational experimentation confirms that the proposed algorithm performs in an effective manner, and it is able to efficiently solve the sets of instances used for the benchmark. Moreover, the performance of the proposed algorithm is compared with a standard genetic algorithm (GA), a scatter search (SS) method and a memetic algorithm (MA). Acceptable results are obtained for all four implemented metaheuristics, but the path homotopy algorithm stands out.Originality/valueA novel metaheuristic is proposed for the first time. It uses topology concepts to design an algorithmic framework to solve binary programming problems in an effective and efficient manner.

Homotopy Theory in Digital Topology

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.