**Advances in Homotopy Theory VI **Workshop

This is the sixth edition of a twice-yearly workshop alternating between the Southampton Centre for Geometry, Topology and Applications (CGTA) and the Beijing Institute of Mathematical Sciences and Applications (BIMSA). The aims are to promote exciting new work in homotopy theory, with an emphasis on younger mathematicians, and to showcase the subject’s wide relevance to other areas of mathematics and science.

**Program**

**2024-06-17 **

**19:00-19:50**

**Daisuke Kishimoto ( Kyushu University )**

**Vector fields on noncompact manifolds**

I will present the Poincare-Hopf theorem for a bounded vector field on a connected noncompact manifold having a cocompact and properly discontinuous action of a discrete group. As a corollary, we will see that every bounded vector field on such a noncompact manifold has infinitely many zeros whenever the orbit manifold has nontrivial Euler characteristic and the acting group is amenable.

**20:00-20:50**

**Shuang Wu ( 吴双 , Beijing Forestry University & BIMSA )**

**Applications of GLMY theory in metabolomic networks of complex diseases**

Human diseases involve metabolic alterations. Metabolomic profiles have served as a biomarker for the early identification of high-risk individuals and disease prevention. However, current approaches can only characterize individual key metabolites, without taking into account their interactions. This work have leveraged a statistical physics model to combine all metabolites into bDSW networks and implement GLMY homology theory to analyze and interpret the topological change of health state from symbiosis to dysbiosis. The application of this model to real data allows us to identify several hub metabolites and their interaction webs, which play a part in the formation of inflammatory bowel diseases.

**21:00-21:50**

**Sandip Samanta ( Indian Institute of Science Education and Research Kolkata )**

**On Generalized Brace Product**

In the 1970s, James introduced the brace product for fibrations admitting a section to study the decomposability of certain fibrations. We have generalized this notion to investigate the H-splitting of based loop space fibrations of a given fibration. First, we will demonstrate that the generalized brace product is the sole obstruction to such splitting. Then, we will explore the connection between the generalized brace product and a generalized notion of Whitehead's J-homomorphism. Additionally, we will present examples in rationalized spaces, where the equivalence of the vanishing of the generalized brace product and the standard brace product simplifies computations. This work is based on our recent preprint, available at the link: https://arxiv.org/abs/2401.16206, co-authored with Dr. Somnath Basu and Dr. Aritra Bhowmick.

**2024-06-18**

**19:00-19:50**

**Sadok Kallel ( American University of Sharjah )**

**The topology of spaces of maps from a Riemann surface to complex projective space**

The space of continuous maps Map(M,N) between two Riemannian manifolds M and N is a fundamental object of study in algebraic topology, more particularly when the source space M is a sphere. We will address the case when M=C is a Riemann surface of positive genus and N is a complex projective n-space. This mapping space has received considerable attention in the literature, by physicists and mathematicians alike. It breaks down into connected components indexed by an integer (the "charge"). We give an overview of the relevant results, and then describe the homology of these components. This is ongoing work with Paolo Salvatore (Rome).

**20:00-20:50 **

**Fedor Vylegzhanin ( Moscow State University )**

**Loop homology of moment-angle complexes in the flag case**

Moment-angle complexes $\mathcal{Z}_{K}$, an important class of CW-spaces with torus action, are parametrized by simplicial complexes $K$. We study their homotopy invariants in the case of flag simplicial complexes. For arbitrary coefficient ring $k$ we describe a presentation of the Pontryagin algebra $H_*(\Omega\mathcal{Z}_K;k)$ by multiplicative generators and relations. Proof uses the connection between presentations of connected graded algebras and the Tor functor.

Applying recent results by Huang, Berglund, Stanton, we prove that such moment-angle complexes are rationally coformal, give a necessary condition for their formality, and compute their homotopy groups in terms of homotopy groups of spheres. If time permits, I will outline similar results for homotopy quotients of moment-angle complexes, including quasitoric manifolds (work in progress).

**21:00-22:30**

**Ilya Alekseev; Vasiliy Ionin ( PDMI RAS & Saint Petersburg University; Chebyshev Laboratory, Saint Petersburg University )**

**Mixing braids, automorphisms, simplicial methods, and homotopy groups of spheres**

We discuss some fundamental connections between low-dimensional topology and homotopy theory. The first part of the talk is devoted to the interplay between braid groups and homotopy groups of spheres. We begin by investigating the impact of some geometric transformations on Brunnian braids and highlighting their non-preserving nature. This analysis leads us to new simplicial structures in braid theory and an action of the automorphism groups $\mathrm{Aut}(P_n)$ of the pure braid groups $P_n$ on homotopy groups of the two-sphere $S^2$. In particular, in the second part of the talk, we construct a simplicial group built on commutator subgroups $[P_n, P_n]$ of the pure braid groups, which relies on the Fulton-MacPherson compactifications of configuration spaces. By inspecting the derived subgroup of J. Milnor's free group construction, we prove that this simplicial group is homotopy equivalent to the three-sphere $S^3$. As an application, we show how this economical model for the three-sphere leads to some interesting Wu-type formulas for the homotopy groups $\pi_n(S^3)$.

**2024-06-19**

**19:00-19:50**

**Juxin Yang ( 杨聚鑫 , BIMSA )**

**The extension problems for three far-unstable 33-stem homotopy groups and Toda brackets of diverse shapes**

Firstly, we will introduce our methods to tackle the extension problems for the homotopy groups $\pi_{39}(S^6)$, $\pi_{40}(S^7)$ and $\pi_{41}(S^8)$ localized at 2, the puzzles having unsolved for forty-five years. Our ability to address these extension problems is largely attributed to the utilization of a rectangular Toda bracket indexed by 2, a Toda bracket of new shape defined by us.

Then we will introduce the Toda bracket (Tbr) in the spirit of Toda's 1962 monograph and its developments, including the 3-fold Tbr, 4-fold Tbr, left and right matrix Tbr, rectangular Tbr and Z-shape Tbr. We shall also introduce the applications we got in recent years, namely, the determinations of some homotopy groups of spheres and SO(n). It is worth mentioning that

“the Toda bracket is an art of constructing homotopy liftings and homotopy extensions of maps, it plays a fundamental role in dealing with composition relations of homotopy classes”

**20:00-20:50**

**Briony Eldridge ( University of Southampton )**

**Loop spaces of polyhedral products associated with substitution complexes**

Polyhedral products are a topological space formed by gluing together ingredient spaces in a manner governed by a simplicial complex. They appear in many areas of study, including toric topology, combinatorics, commutative algebra, complex geometry and geometric group theory. A fundamental problem is to determine how operations on simplicial complexes change the topology of the polyhedral product. In this talk, we consider the substitution complex operation, a special case of the polyhedral join operation. We obtain a description of the loop space associated with some substitution complexes, and build a new family of simplicial complexes such that the homotopy type of the loop space of the moment angle complex is a product of spheres and loops on spheres.

**21:00-21:50**

** Antonio Viruel ( Universidad de Malaga )**

**On the group of self homotopy equivalences of polyhedral product of BG's**

In the context of the Kahn realization problem, we study the group of self-homotopy equivalences of polyhedral products of classifying spaces of simply connected simple compact Lie groups. Specifically, for a given polyhedron K and a simply connected simple compact Lie group G, we describe the group of self-homotopy equivalences of (BG)^K. We demonstrate that this group fits into a short exact sequence involving Aut(K) and Out(G). This is a joint work with Cristina Costoya (USC).