Unstable HomotopyForum II at AMSS

Abstract

The complex projective spaces and Grassmannians have, in a sense, the nicest possible rational models. In the stable category, they are equivalent to a wedge of spheres. Unstably, they are formal, and also their underlying cohomology algebras are intrinsically formal as defined by Halperin and Stasheff. We prove analogous results in the equivariant case. The results feature in joint work with Vanny Doem, Chandal Nahak, and Soumyadip Thandar.

Abstract

Typical enumerative problems in algebraic geometry include finding the \(d\) roots of a generic polynomial in one variable of degree \(d\), finding the \(27\) lines on a smooth cubic surface, and their higher dimensional analogs. We introduce the concept of topological complexity of enumerative problems, a positive integer that measures the least possible number of branches in algorithms that solve an enumerative problem up to an \(\epsilon\) error.

We are interested in lower bounds for the topological complexity of enumerative problems. We introduce finite covering spaces associated to the enumerative problems and the concept of Schwarz genus of a covering space, which produces lower bounds for topological complexity and can be detected by certain morphisms of cohomology rings. Finally, we present lower bounds for three enumerative problems.

This is a joint work with Weiyan Chen.

Abstract

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an \(\infty\)-category \(\mathrm{DGra}_\infty\). Our main result shows that \(\mathrm{DGra}_\infty\) is equivalent to the \(\infty\)-category of spaces.

Abstract

The notion of an \(A_n\)-space in the sense of Stasheff is a space with an \(A_n\)-form, and arises naturally as a homotopy-theoretic relaxation of an associative multiplication. However, its formal definition requires the existence of a strict unit.

Around 2012, it was shown that all inclusions of total spaces in the \(A_n\)-structure constructed from an \(A_n\)-form without strict unit factor through contractible spaces, thereby recovering the expected \(A_n\)-form with a strict unit. On the other hand, Muro and Tonks introduced a version of unital associahedra that weakens the strict unit condition homotopically, in order to model topological analogues of Fukaya's \(A_\infty\)-structures.

In this talk, we present the dual of the simplex category as a basis for constructing a thickened simplex category using Stasheff or Muro-Tonks associahedra.

This approach provides a construction of dual projective spaces for unital co-\(A_n\)-spaces, which are dual to the projective spaces associated with unital \(A_n\)-spaces. In particular, these dual projective spaces are expected, in the limit, to yield desuspensions of co-\(A_\infty\)-spaces.

We also remark that diffeological methods provide a convenient framework for handling such constructions via a blowup diffeology inspired by Kihara.

Abstract

I will talk about joint work with Tsuyoshi Kato and Mitsunobu Tsutaya on the Lefschetz fixed-point theory for noncompact manifolds. We develop the theory in the setting of uniformly continuous self-maps that stay within a bounded distance from the identity map, together with uniformly continuous homotopies. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, and develop an obstruction theory based on it.

Abstract

Let \(p\) be a fixed prime. A discrete \(p\)-toral group is a group that contains a normal subgroup \(T \cong (\mathbb{Z}/p^\infty)^r\) of \(p\)-power index, where \(r \geq 0\). A fusion system over a discrete \(p\)-toral group \(S\) is a category whose objects are the subgroups of \(S\) and whose morphisms are homomorphisms between them satisfying a set of axioms.

Fusion systems over discrete \(p\)-toral groups have been shown to admit a classifying space that is unique up to equivalence. Particular examples arise from finite groups, compact Lie groups, \(p\)-compact groups, and linear torsion groups. In all these cases, the classifying space of the corresponding fusion system is homotopy equivalent to the \(p\)-completed classifying space of the object that gives rise to it.

A fusion system over a finite \(p\)-group is said to be exotic if it does not arise from a genuine finite group. Until recently it was not clear what should be meant by an exotic fusion system over an infinite discrete \(p\)-toral group. The project reported in this talk has two parts.

In the first part, we investigated four different ways of realising fusion systems over discrete \(p\)-toral groups by discrete groups, the most general of which we call sequential realisability. This means that the fusion system in question is, in the appropriate sense, a colimit of a sequence of \(p\)-local finite groups, each of which is realisable by a genuine finite group.

In the second part, we concentrated on a specific very general type of discrete group. A discrete group \(G\) is locally finite if every finitely generated subgroup is finite, a \(p\)-group if every element in it has \(p\)-power order, and artinian if it satisfies the descending chain condition. A group is \(p\)-artinian if every \(p\)-subgroup of it is artinian.

If \(G\) is locally finite and \(p\)-artinian, then every \(p\)-subgroup of \(G\) is locally finite and artinian, hence discrete \(p\)-toral by standard results. We show how to associate a fusion system to any locally finite \(p\)-artinian group \(G\) in such a way that the classifying space of the fusion system is homotopy equivalent to the \(p\)-completed classifying space of \(G\).

In doing so we deal with several nontrivial problems, one of which is that our groups need not have Sylow \(p\)-subgroups in the usual sense. To prove the equivalence of classifying spaces, we use a theorem of Gonzalez expressing the cohomology of the classifying space of a fusion system over a discrete \(p\)-toral group as the stable elements in the cohomology of the Sylow subgroup.

After introducing the basics, I will concentrate on the second part of the project.

Abstract

Knotoid theory is a generalization of knot theory introduced by Turaev in 2012. In recent years, various invariants of knotoids have been studied. In this talk, we mainly discuss unknotting operations and unknotting numbers of plus-welded knotoids. We show that two classical operations, crossing change and crossing virtualization, serve as unknotting operations for plus-welded knotoids. For both operations, by utilizing the descending diagram and the warping degree, we obtain upper bounds for corresponding unknotting numbers of plus-welded knotoids. This is joint work with Andrei Vesnin and Xuan Yang.

Abstract

Bott periodicity states that real topological \(K\)-theory \(KO\) is 8-periodic. In this talk, we establish a minimal periodicity theorem for higher-height analogues of \(KO\) in chromatic homotopy theory.

As an application, at the prime 2, we obtain lower bounds on the exponents of \(v_k\)-self maps of generalized Moore spectra of the form \(S/(2, v_1^{i_1}, \ldots, v_n^{i_n})\). These bounds are optimal in known cases. This is joint work in progress with Zhipeng Duan, Guozhen Wang, and Wei Yang.

Abstract

The classical homotopy theory of digraphs was introduced by Grigor'yan--Lin--Muranov--Yau in 2014. In their framework, the higher homotopy groups of a based digraph are defined recursively through the loop-digraph \(LG^*\):

In this talk, we present a geometric approach to higher homotopy groups of digraphs. We first construct digraph spheres and introduce \(C\)-homotopy, leading to a direct sphere-based definition

We also introduce a simplified loop structure \(\Omega(G)\), which is obtained in two complementary ways: directly as a minimal \(C\)-homotopy model, and from the classical loop-digraph \(LG^*\) through an enlargement and retraction procedure. We prove that these two constructions give exactly the same digraph \(\Omega(G)\), and that \(\Omega(G)\) preserves the fundamental homotopy information of \(LG^*\):

Consequently, we obtain the unified description

Abstract

This work introduces pro-tangles, a natural generalization of classical tangles, which are functors from the Boolean cube to Bar-Natan's cobordism category. By employing the simplicial Yoneda embedding, we construct the Khovanov simplicial presheaf of a pro-tangle as a homotopy colimit and prove that this simplicial presheaf is representable, with representing object the classical Khovanov simplicial object.

We establish a fully faithful embedding showing that the weak equivalence class of this simplicial presheaf is determined by the chain homotopy type of the Khovanov complex. Furthermore, we utilize Boolean cube decompositions to construct an algebraic spectral sequence for pro-tangles. This spectral sequence converges to the total Khovanov homology, and its \(E_1\) page is explicitly expressed in terms of the Khovanov homology of reduced tangles.

This categorical setup yields a functorial interpretation of Reidemeister invariance in terms of morphisms of spectral sequences. By applying the tangle TQFT construction, we study this spectral sequence for Hopf clasps, the fundamental structural building blocks in tangle and link theory. We show that the spectral sequence collapses at the \(E_3\) page, which further specializes to an \(E_2\)-collapse under the restriction to Hopf sums.

Finally, we investigate connected sums of pro-tangles and pro-links. To address the module-action dependencies arising from tensor products in multi-connected sums, we introduce a state-dependent modified tensor operator and prove a structural decomposition theorem that generalizes the classical result at the chain complex level.

Abstract

Classifying equivariant smooth closed manifolds up to equivariant bordism is one of the fundamental problems in topology. We will mainly focus on equivariant geometric unoriented bordism of \(G\)-actions fixing isolated points where \(G=\mathbb{Z}_2^k\), which can be directly associated with \(G\)-representation theory. In this talk, I will introduce some new progress, especially for the homology description and the dimension formulae of equivariant geometric bordism groups, involving a connection with the universal complexes of DJ theory and the uses of matroid theory, spectral sequences, and related tools.

Abstract

In this talk, we first establish a sufficient condition for a simply connected Poincaré complex to be the homotopy type of a closed topological manifold. Then, we construct a family of highly connected Poincaré complexes that are homotopy equivalent to closed topological manifolds, yet not homotopy equivalent to any smooth manifold. Finally, we achieve a complete homotopy classification of closed \(2k\)-connected framed \((4k+2)\)-dimensional manifolds with Kervaire invariant one for \(k=7,15,31\).

Abstract

Toric orbifolds are orbifolds equipped with torus actions analogous to those on toric varieties. Masuda and Suh posed the Cohomological Rigidity Problem, which asks whether their topology or geometry can be determined solely by their cohomology. While affirmative answers are known in several smooth cases, the problem remains largely open in general. In this talk I will present recent results on the cohomological rigidity of toric orbifolds up to homotopy equivalence.

This is based on joint work with Tyrone Cutler, Xin Fu, Jongbaek Song, and Stephen Theriault.

Abstract

A conjecture of Moore asserts a deep connection between the torsion and torsion-free parts of the homotopy groups of any simply connected finite CW complex. This is closely related to a conjecture of Anick, which asserts a connection between the homotopy groups of such spaces and the homotopy groups of spheres.

Much work has been done recently to verify these conjectures in the context of polyhedral products. These are natural subspaces of Cartesian products of spaces indexed by a simplicial complex, and they unify constructions across mathematics.

In this talk, I will summarise the work of Hao, Sun, and Theriault which verified Moore's conjecture for an important class of polyhedral products. I will then discuss work of various authors on Anick's conjecture, culminating in joint work with Vylegzhanin which verifies the conjecture for most polyhedral products.

Abstract

Analogous to diffeological spaces, \(C^0\)-spaces are defined to be concrete sheaves over the Euclidean site with continuous maps. We will discuss the homotopy theory on these spaces, and compare it with the homotopy theory of topological spaces.

Abstract

In this talk, I will give an introduction to Paul Selick and Jie Wu's \(A^{\min}\) theory for the loop-suspension functor \(X \mapsto \Omega\Sigma X\). The theory gives a functorial decomposition of \(\Omega\Sigma X\) closely related to the tensor coalgebra from the Bott-Samelson theorem. I will explain the main construction and discuss some recent applications in unstable homotopy theory.

Abstract

The Stanley-Reisner rings of simplicial complexes provide the central link between combinatorics and commutative algebra. For a simplicial complex \(K\) and a field \(\mathbb{F}\), the Betti numbers of the Stanley-Reisner ring \(\mathbb{F}[K]\) are important combinatorial invariants of \(K\) that have been intensively studied.

It follows from the work of Evans and Griffith on the Buchsbaum-Eisenbud-Horrocks conjecture that the \(i\)-th Betti number \(\beta^{-i}(\mathbb{F}[K])\) of \(\mathbb{F}[K]\) always satisfies

where \(m\) is the number of vertices of \(K\), and \(\mathrm{mdim}(K)\) is the minimal dimension of the maximal simplices of \(K\). In this talk, we will discuss various properties of those simplicial complexes \(K\) whose \(\beta^{-i}(\mathbb{F}[K])\) reaches the above lower bound for all \(i\).

Especially, we find that the Stanley-Reisner rings of such kind of simplicial complexes are exactly those with minimal Taylor resolutions. Moreover, we will explain how to construct such kind of simplicial complexes inductively from examples with fewer vertices.

Abstract

Representation graded Bredon \(G\)-equivariant cohomology theories are represented by genuine \(G\)-spectra. We introduce the concept of \(R\)-Eulerian sequences for an equivariant commutative ring spectrum \(R\). Each \(R\)-Eulerian sequence corresponds to a stable \(R\)-cohomology operation.

We apply our theory to equivariant ordinary cohomology to produce genuine equivariant lifts of the classical Steenrod operations for all finite groups. This is joint work with Prasit Bhattacharya, Alex Waugh, and Mingcong Zeng.