北京雁栖湖应用数学研究院

The unit octonions, $S^7$, is an H-space which is not a Lie group due to failure of associativity. We examine the extent to which $S^7$ has analogies of Lie group concepts such as maximal torus, Weyl group, Lie algebra, and exponential map. Moreover, we present a method for calculating the homology of the space of commuting n-tuples in $S^7$ by induction on $n$.

The Lusternik-Schnirelmann category (LS-category) is a numerical homotopy invariant of topological spaces, measuring the minimal number of contractible open sets required to cover the space. For a fibration, the LS-category can be bounded above by the categories of its base and fiber. In rational homotopy theory, certain fibrations are modeled by relative Sullivan algebras, for which an algebraic analogue of the LS-category is defined. Felix, Halperin, and Thomas posed the question of whether the LS-category of a relative Sullivan algebra is similarly bounded by the categories of its base algebra and fiber algebra. In this talk, we provide a positive answer to this question.

One of the classical applications of Sullivan’s rational homotopy theory was the result that the components $\pi_0aut(X)$ of the space $aut(X)$ of self-homotopy equivalences of a simply connected finite complex $X$ form an arithmetic group. I will talk about work that, roughly speaking, is about extending such $\pi_0$-statements to space-level statements. In joint work with Tomas Zeman, we construct an algebraic model for the classifying space $Baut(X)$. The main application is a formula for the rational cohomology ring of $Baut(X)$ in terms of the cohomology of arithmetic groups and dg Lie algebras. In more recent joint work with Robin Stoll, we establish similar results for classifying spaces of tangential homotopy equivalences and block diffeomorphisms of smooth manifolds with boundary.

Virtual links are links embedded in thickened surfaces. Their groups correspond to the fundamental groups of their complement spaces within these thickened surfaces. In this talk, we demonstrate that for a virtual link $L$ with supporting genus $g > 0$, the complement space of $L$ is irreducible and forms a $K(\pi, 1)$-space. Furthermore, we establish that under certain hypotheses, there is an isomorphism between the intersecting subgroup and the symmetric commutator subgroup of all normal closures of meridians in the virtual link group. This is based on joint work with Ruzhi Song.

Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory provide theoretical tools to explore the complexity and entanglement of curves in 3-space. However, classical knot theory focuses on global topological properties and lacks the consideration of local structural information, which is critical in practical applications. In this work, two localized models based on the Jones polynomial were proposed, namely, the multi-scale Jones polynomial and the persistent Jones polynomial. The stability of these models, especially the insensitivity of the multi-scale and persistent Jones polynomial models to small perturbations in curve collections, was analyzed, thus ensuring their robustness for real-world applications.

The path homology introduced by Grigor’yan, Lin, Muranov and Yau plays a central role in digraph topology and the emerging field of GLMY theory more generally. Unfortunately, the computation of the path homology of a digraph is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In particular, our understanding of the path chains is the primary obstruction to the development of fast path homology algorithms, which in turn would enable the practicality of a wide range of applications to directed networks.<br>I will introduce an inductive method of constructing elements of the path homology chain modules from elements in the proceeding two dimensions. When the coefficient ring has prime characteristic the inductive elements generate the path chains. Moreover, in low dimensions the inductive elements coincide with naturally occurring generating sets up to sign, making them excellent candidates to reduce to a basis.<br>Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph. During the talk I will demonstrate how inductive elements yield the explicit structure of the dimension 3 path chains and enable the construction of a sequence of digraphs whose path Euler characteristic can<br>differ arbitrarily depending on the choice of coefficients.

This is joint work with Guille Carrión Santiago and Ramón Flores. Quillen’s plus construction kills the maximal perfect subgroup without changing the integral homology and there are analogous constructions for other homology theories. We focus in this talk on rational homology and construct an explicit model of a universal HQ-acyclic space, so that the corresponding nullification functor provides a functorial plus construction for ordinary homology with rational coefficients. Motivated by classical results about Quillen's plus construction for integral homology, we use this description to study the HQ-acyclization functor and the associated acyclization-plus construction fiber sequence.

The study of exotic smooth structures on manifolds is a central problem in differential topology. In particular, the classification of smooth structures on a given smooth manifold is closely related to the <i>concordance inertia group</i>, a subgroup of the group of homotopy spheres. In this talk, I will compute the concordance inertia group of the product $M\times \mathbb{S}^k$, where M is a simply connected, closed, smooth 6-manifold and $1 \le k \le 10$. The approach involves analysing the top cell attaching map of M stably, together with known computations of the stable homotopy groups of spheres.<br>In addition, I will discuss the concordance smooth structure sets $\mathcal{C}(M\times \mathbb{S}^1)$ and $\mathcal{C}(\mathbb{C}P^3 \times \mathbb{S}^k)$ for all $2 \le k \le 10$. As an application, I will present a complete diffeomorphism classification of smooth manifolds that are homeomorphic to $\mathbb{C}P^3 \times \mathbb{S}^k$ for $1 \le k \le 7$, and outline the proof of the classification for one particular value of $k$.

Polyhedral products are formed by combinatorially gluing Cartesian products of spaces according to associated simplicial complexes. They recover many important constructions including Davis-Janskiewic spaces, moment-angle complexes, and classifying spaces of right-angled Artin groups. Recently, we introduced weighted polyhedral products as a generalization of polyhedral products, and used them to study the realization problem for algebras related to products of spheres. Moreover, we explored some of their homotopy theoretic properties and computed their cohomology in special cases. This is joint work with Donald Stanley and Stephen Theriault.