arXiv subjects

Yichen Tong

Publications and source records attributed to Yichen Tong.

7 recordsLinked to original sources

arXiv2026

Discrete homotopy groups of cubical sets

We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. Here, quasisymmetric cubical sets are cubical sets equipped with coordinate permutation symmetries that are compatible with faces and degeneracies, but not necessarily with connections. We give a purely combinatorial construction of the left adjoint of the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, and prove that the unit of this adjunction is an objectwise weak equivalence. As a consequence, we obtain a purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets. As an application, we establish the Hurewicz theorem for the discrete homotopy groups of quasisymmetric cubical sets.

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arXiv2024

The fundamental group and the magnitude-path spectral sequence of a directed graph

The fundamental group of a directed graph admits a natural sequence of quotient groups called $r$-fundamental groups, and the $r$-fundamental groups can capture properties of a directed graph that the fundamental group cannot capture. The fundamental group of a directed graph is related to path homology through the Hurewicz theorem. The magnitude-path spectral sequence connects magnitude homology and path homology of a directed graph, and it may be thought of as a sequence of homology of a directed graph, including path homology. In this paper, we study relations of the $r$-fundamental groups and the magnitude-path spectral sequence through the Hurewicz theorem and the Seifert-van Kampen theorem.

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arXiv2024

Homotopy commutativity in quasitoric manifolds

We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is a product of $3$-simplices $(\Delta^3)^n$ and the characteristic matrix is equivalent to a matrix of certain type. Quasitoric manifolds over $(\Delta^3)^n$ include generalized Bott manifolds, and we also construct an infinite family of homotopy nonequivalent generalized Bott manifolds over $(\Delta^3)^n$, only half of them have homotopy commutative loop spaces. In particular, for each $n\ge 2$, there are infinitely many homotopy types in $6n$-dimensional quasitoric manifolds having homotopy (non)commutative loop spaces.

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arXiv2023

Homotopy commutativity in symmetric spaces

We extend the former results of Ganea and the two of the authors with Takeda on the homotopy commutativity of the loop spaces of Hermitian symmetric spaces such that the loop spaces of all irreducible symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative.

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arXiv2022

Self-Closeness Numbers of Rational Mapping Spaces

For a closed connected oriented manifold $M$ of dimension $2n$, it was proved by M\o ller and Raussen that the components of the mapping space from $M$ to $S^{2n}$ have exactly two different rational homotopy types. However, since this result was proved by the algebraic models for the components, it is unclear whether other homotopy invariants distinguish their rational homotopy types or not. The self-closeness number of a connected CW complex is the least integer $k$ such that any of its self-map inducing an isomorphism in $\pi_*$ for $*\le k$ is a homotopy equivalence, and there is no result on the components of mapping spaces so far. For a rational Poincar\'e complex $X$ of dimension $2n$ with finite $\pi_1$, we completely determine the self-closeness numbers of the rationalized components of the mapping space from $X$ to $S^{2n}$ by using their Brown-Szczarba models. As a corollary, we show that the self-closeness number does distinguish the rational homotopy types of the components. Since a closed connected oriented manifold is a rational Poincar\'e complex, our result partially generalizes that of M\o ller and Raussen.

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arXiv2022

Homotopy commutativity in Hermitian symmetric spaces

Ganea proved that the loop space of $\mathbb{C}P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency of the loop spaces of flag manifolds.

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arXiv2021

Self-Closeness Number of Non-Simply-Connected Spaces

The self-closeness number $N\mathcal{E}(X)$ of a space $X$ is the least integer $k$ such that any self-map is a homotopy equivalence whenever it is an isomorphism in the $n$-th homotopy group for each $n\le k$. We discuss the self-closeness numbers of certain non-simply-connected $X$ in this paper. As a result, we give conditions for $X$ such that $N\mathcal{E}(X)=N\mathcal{E}(\widetilde{X})$, where $\widetilde{X}$ is the universal covering space of $X$.

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