The Lusternik-Schnirelmann category of 𝑆𝑝(3)

Suárez, Lucía Fernández; Gómez-Tato, Antonio; Strom, Jeffrey; Tanré, Daniel

doi:10.1090/s0002-9939-03-07019-9

Cited by 17 publications

(18 citation statements)

References 25 publications

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“…We use a sequence computation to give a simple alternative proof of this result. Proof According to Fernández-Suárez, Gómez-Tato, Strom and Tanré [14], and Iwase and Mimura [23], wcat(Sp(3)) = cat(Sp(3)) = 5. Since weak category is a genus invariant, we have cat(X) ≥ wcat(X) = wcat(Sp(3)) = 5…”

Section: Conjecturementioning

confidence: 99%

“…However, H * (G 2 ; A) = 0 for * = 12, 13 and any abelian group A, so σ G 2 (4) = 12, 13 by Theorem 3.4(b). We conclude that σ G 2 = (0, 3,6,9,14). cat(X) ≤ dimension(X) connectivity(X) In [16], Ganea generalized this familiar upper bound to obtain an upper bound for the category of X in terms of the set of dimensions in which H * (X) is nontrivial.…”

Section: O Omentioning

confidence: 99%

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Categorical sequences

Nendorf

Scoville

Strom

2006

Algebr. Geom. Topol.

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We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skeleta. The k th term in the categorical sequence of a CW complex X , X .k/, is the least integer n for which cat X .X n / k . We show that X is a well-defined homotopy invariant of X . We prove that X .kCl/ X .k/C X .l/, which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp.3/, then cat.X / D cat.Sp.3//. 55M30; 55P62 IntroductionThe Lusternik-Schnirelmann category of a topological space X is the least integer k for which X has an open cover X D X 0 [ X 1 [ [ X k with the property that each inclusion map X j ,! X is homotopic to a constant map; it is denoted cat.X /. This homotopy invariant of topological spaces was first introduced by Lusternik and Schnirelmann in 1934 as a tool to use in studying functions on (compact) manifolds: a smooth function f W M ! ‫ޒ‬ must have at least cat.M / C 1 critical points.If X is a CW complex, then X n D X n 1 [˛.n cells/, and therefore cat.X n / Ä cat.X n 1 / C 1. Berstein and Hilton asked [3] what conditions must be placed on the attaching map˛in order to guarantee that equality holds in this upper bound; the answer is that equality holds when a certain set of generalized Hopf invariants does This is done via the categorical sequence of a space X , which is a function X W ‫ގ‬ ! ‫ގ‬ [ f1g defined byX n / kg; where cat X .X n / is the category of X n relative to X (see Definition 4) 1 . It is shown in Propositions 2.1 and 2.2 that X is a well-defined homotopy invariant of X ; ie, when n is larger than the connectivity of X , cat X .X n / depends only on n and the homotopy type of X , and not on any choices made in constructing a CW decomposition of X . If X is finite-dimensional, then X determines cat.X /; examples due to Roitberg [28] show that this is not true for infinite-dimensional spaces. In any case, the categorical sequence of X holds a wealth of useful information.Though we are not directly concerned with the applications of Lusternik-Schnirelmann category to critical point theory in this paper, categorical sequences could play a useful role there. For example, in the study of the n-body problem, one is often interested in infinite-dimensional Sobolev spaces W ; in order to apply the Lusternik-Schnirelmann method in thi...

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Section: Conjecturementioning

confidence: 99%

Section: O Omentioning

confidence: 99%

Categorical sequences

Nendorf

Scoville

Strom

2006

Algebr. Geom. Topol.

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“…James and Singhof [16] using explicit cone decompositions of Spin (7) and SU (4). Then the Ganea conjecture on L-S category holds for all these Lie groups, since the L-S and the strong L-S categories are equal to the cup-length:…”

Section: Introductionmentioning

confidence: 98%

Lusternik–Schnirelmann category of non-simply connected compact simple Lie groups

Iwase

Mimura

Nishimoto

2005

Topology and its Applications

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Let F → X → B be a fibre bundle with structure group G, where B is (d − 1)-connected and of finite dimension, d 1. We prove that the strong L-S category of X is less than or equal to m+ dim B d , if F has a cone decomposition of length m under a compatibility condition with the action of G on F . This gives a consistent prospect to determine the L-S category of non-simply connected Lie groups. For example, we obtain cat(PU(n)) 3(n − 1) for all n 1, which might be best possible, since we have cat(PU(p r )) = 3(p r − 1) for any prime p and r 1. Similarly, we obtain the L-S category of SO(n) for n 9 and PO(8). We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.  2004 Elsevier B.V. All rights reserved. MSC: primary 55M30; secondary 22E20, 57N60

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“…This homotopy invariant is often difficult to compute, particularly in the context of spaces of quaternionic matrices. For instance, in the case of the symplectic groups Sp(n), we only know some low values, such as cat Sp(2) = 3 [18], cat Sp(3) = 5 [4], and some bounds, such as cat Sp(n) n + 2 when n 3 [8] or cat Sp(n)…”

Section: Introductionmentioning

confidence: 99%