[Paper Review] On Lusternik-Schnirelmann category of SO(10)

Norio Iwase, Kai Kikuchi+1 more|arXiv (Cornell University)|Dec 21, 2007

Homotopy and Cohomology in Algebraic Topology9 references5 citations

TL;DR

This paper establishes a new homotopical criterion for bounding the Lusternik-Schnirelmann (L-S) category of a principal bundle $\mathrm{SO}(10) \to S^9$ with structure group $\mathrm{SO}(9)$, using a cone-decomposition of $\mathrm{SO}(9)$ and conditions on the characteristic map and Berstein-Hilton Hopf invariant. The key result is that $\operatorname{cat}(\mathrm{SO}(10)) = 21$, matching its mod 2 cup-length, supporting a conjecture that this equality holds for all $\mathrm{SO}(n)$.

ABSTRACT

Let $G$ be a compact connected Lie group and $p : E o ΣA$ be a principal G-bundle with a characteristic map $α: A o G$, where $A=ΣA_{0}$ for some $A_{0}$. Let $\{K_{i}{ o} F_{i-1}{\hookrightarrow} F_{i} \,|\, 1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} Σ{K_{1}} \; ext{and}\; F_{n}{\simeq} G \}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=Σ{K'_{1}} \subset F_{1}$ with $K'_{1} \subset K_{1}$ which satisfy $F_{i}F'_{1} \subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\operatorname{cat}(X) \le m{+}1$, if firstly the characteristic map $α$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(α)$ vanishes in $[A, ΩF'_1{\ast}ΩF'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10) o S^{9}$ to determine L-S category of $\mathrm{SO}(10)$.

Motivation & Objective

• To develop a homotopical criterion for bounding the L-S category of a principal bundle $E \to \Sigma A$ with structure group $G$, based on cone-decompositions and characteristic maps.
• To address the open problem of computing $\operatorname{cat}(\mathrm{SO}(10))$, a key Lie group in geometric topology.
• To prove that $\operatorname{cat}(\mathrm{SO}(10)) = 21$, matching its mod 2 cup-length, suggesting a general pattern for $\mathrm{SO}(n)$.

Proposed method

• Construct a cone-decomposition of $\mathrm{SO}(9)$ of length 20, with $F_0 = \{\ast\}$, $F_1 \simeq \Sigma \mathbb{C}P^3$, and $F_{20} \simeq \mathrm{SO}(9)$, where $F_i$ are built via cellular attachments.
• Define $F'_1 = \Sigma \mathbb{C}P^3 \subset F_1$ and verify that the multiplication $\mu|_{F_i \times F'_1}$ is compressible into $F_{i+1}$ up to homotopy for $1 \leq i < 20$, ensuring compatibility with the group structure.
• Show that the characteristic map $\alpha: S^8 \to \mathrm{SO}(9)$ is compressible into $F'_1 = \Sigma \mathbb{C}P^3$, using the fact that $\alpha$ is homotopic to the suspension of the canonical $S^1$-bundle over $\mathbb{C}P^3$.
• Prove that the Berstein-Hilton Hopf invariant $H_1(\alpha)$ vanishes in $[S^8, \Omega F'_1 * \Omega F'_1]$, due to $\alpha$ being a suspension map.
• Apply Theorem 1.4, which gives $\operatorname{cat}(E) \leq m+1$ under three conditions: compressibility of $\alpha$ into $F'_1$, vanishing of $H_1(\alpha)$, and $K_m$ being a sphere, to conclude $\operatorname{cat}(\mathrm{SO}(10)) \leq 21$.
• Combine the upper bound with the known lower bound $\operatorname{cup}(\mathrm{SO}(10); \mathbb{F}_2) = 21$ to conclude equality.

Experimental results

Research questions

• RQ1 What is the Lusternik-Schnirelmann category of $\mathrm{SO}(10)$, and does it equal its mod 2 cup-length?
• RQ2 Can the cone-decomposition method be used to compute $\operatorname{cat}(\mathrm{SO}(10))$ via the structure of the principal bundle $\mathrm{SO}(9) \to \mathrm{SO}(10) \to S^9$?
• RQ3 Under what homotopical conditions does the Berstein-Hilton Hopf invariant and cone-decomposition structure ensure $\operatorname{cat}(E) \leq m+1$ for a principal bundle $E \to \Sigma A$?

Key findings

• The Lusternik-Schnirelmann category of $\mathrm{SO}(10)$ is exactly 21, as established by combining an upper bound from a cone-decomposition and a lower bound from cup-length.
• The upper bound $\operatorname{cat}(\mathrm{SO}(10)) \leq 21$ is derived from Theorem 1.4, which applies when the characteristic map $\alpha$ is compressible into $F'_1 = \Sigma \mathbb{C}P^3$, the Hopf invariant $H_1(\alpha)$ vanishes, and the top cell $K_{20}$ is a sphere.
• The characteristic map $\alpha: S^8 \to \mathrm{SO}(9)$ is homotopic to the suspension of the canonical $S^1$-bundle over $\mathbb{C}P^3$, ensuring its compressibility into $F'_1 = \Sigma \mathbb{C}P^3 \subset F_1$.
• The Berstein-Hilton Hopf invariant $H_1(\alpha)$ vanishes because $\alpha$ is a suspension map, which implies triviality in the relevant homotopy group.
• The cone-decomposition of $\mathrm{SO}(9)$ has length 20, with $F_{20} \simeq \mathrm{SO}(9)$, and the top cell $K_{20} = S^{35}$, satisfying the sphere condition in Theorem 1.4.
• The result $\operatorname{cat}(\mathrm{SO}(10)) = 21 = \operatorname{cup}(\mathrm{SO}(10); \mathbb{F}_2)$ supports the conjecture that $\operatorname{cat}(\mathrm{SO}(n)) = \operatorname{cup}(\mathrm{SO}(n); \mathbb{F}_2)$ for all $n$.