On Lusternik–Schnirelmann category of ${\bf SO}(10)$

Abstract

Let G be a compact connected Lie group and p : E → ΣA be a principal G-bundle with a characteristic map α :up to homotopy for any i. Our main result is as follows: we have cat(X) ≤ 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 * ΩF 1 ] and thirdly K m is a sphere. We apply this to the principal bundle SO(9) → SO(10) → S 9 to determine L-S category of SO(10).Let R be a commutative ring and X a connected space. The cup-length of X with coefficients in R is the least non-negative integer k (or ∞) such that

An explicit formula for the cup-length of the rotation group

“…Due to I. James and W. Singhof [5], N. Iwase, M. Mimura, and T. Nishimoto [3], and N. Iwase, K. Kikuchi, and T. Miyauchi [4], it is known that this lower bound is sharp for n = 1, 2, . .…”

“…Of course, our formula for cup(SO(n)) (Theorem 1.1) is of interest in its own right. But it also enables us to transform the conjecture worded in [4], "this would suggest that cat(SO(n)) = cup(SO(n)) for all n," into the following explicit problem.…”

Oriented robot motion planning in Riemannian manifolds

“…This cohomology ring has a particular significance for our computations because of its relation to Lusternik-Schnirelmann category. The values of cat(SO(n)) have been computed for n ≤ 5 by I. James and W. Singhof in [JS99], for n ≤ 9 by N. Iwase, M. Mimura and T. Nishimoto in [IMN05] and for n = 10 by Iwase, K. Kikuchi and T. Miyauchi in [IKM16]. It was shown that for each n ≤ 10, it holds that (4) cat(SO(n)) = cl F 2 (SO(n)) + 1.…”