Homotopy Theory Research Articles

Topological symmetry in quantum field theory

We introduce a definition and framework for internal topological symmetries in quantum field theory, including “noninvertible symmetries” and “categorical symmetries”. We outline a calculus of topological defects which takes advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called “gauging” and “condensation defects”), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.

Topological Interactions Between Homotopy and Dehn Twist Varieties

The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, orientations of twists and the formation of pointed based space under a homeomorphic continuous function. It is shown that the Dehn twisted homotopy under non-retraction admits a left lifting property (LLP) through the local homeomorphism. The LLP extends the principles of Hurewicz fibration by avoiding pullback. Moreover, this paper illustrates that the Dehn twisted homotopy up to a base point in a based space can be formed by considering retraction. As a result, two disjoint continuous functions become point-wise continuous at the base point under retracted homotopy twists. Interestingly, the oriented Dehn twists of a pre-twisted space under homotopy retraction mutually commute in a contractible space.

Examples and cofibrant generation of effective Kan fibrations

We will make two contributions to the theory of effective Kan fibrations, which are a more explicit version of the notion of a Kan fibration, a notion which plays a fundamental role in simplicial homotopy theory. We will show that simplicial Malcev algebras are effective Kan complexes and that the effective Kan fibrations can be seen as the right class in an algebraic weak factorization system. In addition, we will introduce two strengthenings of the notion of an effective Kan fibration, the symmetric effective and degenerate-preferring Kan fibrations, and show that the previous results hold for these strengthenings as well.

The Adams isomorphism revisited

We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more generally in the settings of G-Mackey functors, G-global homotopy theory, and equivariant Kasparov categories.

On Some Applications of P-Contraction Fixed Point Theorems in Bipolar Metric Spaces

In this paper, we have discussed the P-contraction type fixed point theorems of covariant mappings in bipolar metric spaces, and we have shown an example which supports our results. Also we discussed applications to integral equations and Homotopy theory.

Deformations and Homotopy Theory for Rota-Baxter Family Algebras

Deformations and Homotopy Theory for Rota-Baxter Family Algebras

The surjection property and computable type

We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the ϵ-surjection property, that were recently introduced to characterize the notion of computable type arising from computability theory. For a class of spaces including the finite simplicial complexes, we develop techniques to prove or disprove these properties using homotopy and homology theories, and give applications of these results. In particular, we answer an open question on the computable type property, showing that it is not preserved by taking products. We also observe that computable type is decidable for finite simplicial complexes.

Homotopy Theoretic Properties Of Open Books

ABSTRACT We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions we prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

Homotopy, symmetry, and non-Hermitian band topology

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ( PT ) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in PT -symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous PT symmetry breaking is captured by Chern–Euler and Chern–Stiefel–Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.

Binomial rings and homotopy theory

Binomial rings and homotopy theory

Parametrised Presentability Over Orbital Categories

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.

Homotopy in Exact Categories

In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories, and use these to study model structures on categories of chain complexes C h ∗ ( E ) Ch_{*}(\mathcal {E}) which are induced by cotorsion pairs on E \mathcal {E} . As a special case we show that under very general conditions the categories C h + ( E ) Ch_{+}(\mathcal {E}) , C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) , and C h ( E ) Ch(\mathcal {E}) are equipped with the projective model structure, and that a generalisation of the Dold-Kan correspondence holds. We also establish conditions under which categories of filtered objects in exact categories are equipped with natural model structures. When E \mathcal {E} is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. In particular we provide conditions under which C h ( E ) Ch(\mathcal {E}) and C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) are homotopical algebra contexts, thus making them suitable settings for derived geometry.

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

abstract: In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.

KK- and E-theory via homotopy theory

KK- and E-theory via homotopy theory

Low-energy Earth–Moon transfers via Theory of Functional Connections and homotopy

Numerous missions leverage the weak stability boundary in the Earth–Moon–Sun system to achieve a safe and cost-effective access to the lunar environment. These transfers are envisaged to play a significant role in upcoming missions. This paper proposes a novel method to design low-energy transfers by combining the recent Theory of Functional Connections with a homotopic continuation approach. Planar patched transfer legs within the Earth–Moon and Sun–Earth systems are continued into higher-fidelity models. Eventually, the full Earth–Moon transfer is adjusted to conform to the dynamics of the planar Earth–Moon Sun-perturbed, bi-circular restricted four-body problem. The novelty lies in the avoidance of any propagation during the continuation process and final convergence. This formulation is beneficial when an extensive grid search is performed, automatically generating over 2000 low-energy transfers. Subsequently, these are optimized through a standard direct transcription and multiple shooting algorithm. This work illustrates that two-impulse low-energy transfers modeled in chaotic dynamic environments can be effectively formulated in Theory of Functional Connections, hence simplifying their overall design process. Moreover, its synergy with a homotopic continuation approach is demonstrated.

Motivic homotopy theory of algebraic stacks

Motivic homotopy theory of algebraic stacks

Topological aspects of brane fields: Solitons and higher-form symmetries

In this note, we classify topological solitons of n-brane fields, which are nonlocal fields that describe n-dimensional extended objects. We consider a class of n-brane fields that formally define a homomorphism from the n-fold loop space \Omega^n X_DΩnXD of spacetime X_DXD to a space Ε_nΕn. Examples of such n-brane fields are Wilson operators in n-form gauge theories. The solitons are singularities of the n-brane field, and we classify them using the homotopy theory of E_nEn-algebras. We find that the classification of codimension {k+1}k+1 topological solitons with {k≥ n}k≥n can be understood using homotopy groups of \mathcal{E}_nℰn. In particular, they are classified by {\pi_{k-n}(\mathcal{E}_n)}πk−n(ℰn) when {n>1}n>1 and by {\pi_{k-n}(\mathcal{E}_n)}πk−n(ℰn) modulo a {\pi_{1-n}(\mathcal{E}_n)}π1−n(ℰn) action when {n=0}n=0 or {1}1. However, for {n>2}n>2, their classification goes beyond the homotopy groups of \mathcal{E}_nℰn when k < n, which we explore through examples. We compare this classification to n-form \mathcal{E}_nℰn gauge theory. We then apply this classification and consider an {n}n-form symmetry described by the abelian group {G^{(n)}}G(n) that is spontaneously broken to {H^{(n)}\subset G^{(n)}}H(n)⊂G(n), for which the order parameter characterizing this symmetry breaking pattern is an {n}n-brane field with target space {\mathcal{E}_n = G^{(n)}/H^{(n)}}ℰn=G(n)/H(n). We discuss this classification in the context of many examples, both with and without ’t Hooft anomalies.

Stratified homotopy theory of topological ∞-stacks: A toolbox

We exploit the theory of ∞-stacks to provide some basic definitions and calculational tools regarding stratified homotopy theory of stratified topological stacks.

Stratified Noncommutative Geometry

We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable ∞ \infty -categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as E n \mathbb {E}_n -monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group G G , we give a symmetric monoidal stratification of genuine G G -spectra. In the case that G G is finite, this expresses genuine G G -spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory.

Homotopy Theory

The workshop brought together experts in homotopy theory from many areas, including chromatic homotopy theory, algebraic K-theory, derived algebra and equivariant homotopy theory. We had a lecture series on the recent disproof of the telescope conjecture. In addition to a total of 24 research talks we had two gong-shows where participants presented their research in 10-minute talks.