Ganea Conjecture Research Articles

A study of the Ganea conjecture for topological complexity by using weak topological complexity

Abstract In this paper, we provide sufficient conditions for a space X to satisfy the Ganea conjecture for topological complexity. To achieve this, we employ two auxiliary invariants: weak topological complexity in the sense of Berstein–Hilton, along with a certain stable version of it. Several examples are discussed.

Generating functions and topological complexity

We examine the rationality conjecture raised in [1] which states that (a) the formal power series ∑r≥1TCr+1(X)⋅xr represents a rational function of x with a single pole of order 2 at x=1 and (b) the leading coefficient of the pole equals cat(X). Here X is a finite CW-complex and for r≥2 the symbol TCr(X) denotes its r-th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.

The Ganea conjecture for rational approximations of sectional category

We give bounds for the module sectional category of products of maps which generalise a theorem of Jessup for Lusternik–Schnirelmann category. We deduce also a proof of a Ganea type conjecture for topological complexity. This is a first step towards proving the Ganea conjecture for topological complexity in the rational context.

ON n-DEFICIENCY OF GROUPS

Let G be a group of type Fn with n ≥ 2, i.e. G admits a finite n-dimensional (n - 1)-aspherical CW-complex X such that π1(X) is isomorphic to G. We introduce the notion of the n-deficiency dn(G) which is a generalized notion of deficiency of G and present various results about the relation between dn(G) and the L2-Betti numbers of G. We also give some partial results about the Eilenberg–Ganea conjecture and the Whitehead conjecture. These results correspond to a generalization of several earlier results [Quart. J. Math. Oxford (2)38 (1987) 35–44; J. Pure Appl. Algebra7 (1976) 241–248; Quart. J. Math. Oxford (2)32 (1981) 45–55; 2-Knots and Their Groups (Cambridge University Press, 1989)].

A minimum dimensional counterexample to Ganea's conjecture

A 7-dimensional CW-complex having Lusternik–Schnirelmann category equal to 2 is constructed. Using a divisibility phenomenon for Hopf invariants, it is proved that the Cartesian product of the constructed complex with a sphere of sufficiently large dimension also has category 2. This space hence constitutes the minimum dimensional known counterexample to Ganea's conjecture on the Lusternik–Schnirelmann category of spaces.

The Ganea conjecture in proper homotopy via exterior homotopy theory

AbstractIn this article we provide sufficient conditions on a spaceXto verify Ganea conjecture with respect to exterior and proper Lusternik–Schnirelmann category. For this aim we previously develop an exterior version of the Whitehead, cellular approximation, CW-approximation and Blakers–Massey theorems within a homotopy theory of exterior CW-complexes and study their corresponding analogues and consequences in the proper setting.

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

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.

Lusternik–Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik–Schnirelmann (L–S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L–S category as its ‘once punctured submanifold’ N\\{P}, P∈N . Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described.

A∞-method in Lusternik–Schnirelmann category

To clarify the method behind (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634), a generalisation of Berstein–Hilton Hopf invariants is defined as ‘higher Hopf invariants’. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634) for all the cases except the case when p=2. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold M is found not to satisfy Ganea's conjecture on LS category and another closed manifold N is found to have the same LS category as its ‘punctured submanifold’ N−{P}, P∈N . But all examples obtained here support the conjecture in (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623–634).

Fibrewise suspension and Lusternik–Schnirelmann category

Since Iwase disproved the Ganea conjecture the question became to find a characterization of the spaces X which satisfy the Ganea conjecture, i.e. for which the equality cat(X×S k)= cat X+1 holds for any k⩾1. Recently Scheerer et al. (H. Scheerer, D. Stanley, D. Tanré, Fibrewise localization applied to Lusternik–Schnirelmann category, Israel J. Math. (2002) to appear.) have introduced an approximation of the category, denoted by Q cat , and have conjectured that, for a CW-complex X of finite dimension, we have cat(X×S k)= cat X+1 for any k⩾1 if and only if Q cat X= cat X . In this paper, we establish the formula Q cat (X×S k)=Q cat X+1 and deduce from this that if Q cat X= cat X then X satisfies the Ganea conjecture. In other words, a first direction of the conjecture of Scheerer et al. (H. Scheerer, D. Stanley, D. Tanré, Fibrewise localization applied to Lusternik–Schnirelmann category, Israel J. Math. (2002) to appear.) is proved. Using this new sufficient condition for a space to satisfy the Ganea conjecture, we prove that any ( r−1)-connected CW-complex X with r cat(X)⩾3 and dim(X)⩽2r cat(X)−3 satisfies the Ganea conjecture. This shows for example that the Lie group Sp(3) satisfies the Ganea conjecture.

Co-H-spaces and the Ganea conjecture

A non-simply connected co-H-space X is, up to homotopy, the total space of a fibrewise-simply connected pointed fibrewise co-Hopf fibrant j : X→Bπ 1(X) , which is a space with a co-action of Bπ 1( X) along j. We construct its homology decomposition, which yields a simple construction of its fibrewise localisation. Our main result is the construction of a series of co-H-spaces, each of which cannot be split into a one-point-sum of a simply connected space and a bunch of circles, thus disproving the Ganea conjecture.

Suspension of Ganea fibrations and a Hopf invariant

We introduce a sequence of numerical homotopy invariants σ icat , i∈ N , which are lower bounds for the Lusternik–Schnirelmann category of a topological space X . We characterize, with dimension restrictions, the behaviour of σ icat with respect to a cell attachment by means of a Hopf invariant. Furthermore we establish for σ icat a product formula and deduce a sufficient condition, in terms of the Hopf invariant, for a space X∪e p+1 to satisfy the Ganea conjecture, i.e., cat ((X∪e p+1)×S m)=cat (X∪e p+1)+1 . This extends a recent result of Strom and a concrete example of this extension is given.

Bestvina–Brady groups and the plus construction

A recent result of Bestvina and Brady [1, theorem 8·7], shows that one of two outstanding questions has a negative answer; either there exists a group of cohomological dimension 2 and geometric dimension 3 (a counterexample to the Eilenberg–Ganea Conjecture [4]), or there exists a nonaspherical subcomplex of an aspherical 2-complex (a counterexample to the Whitehead Conjecture [11]). More precisely, Bestvina and Brady construct a family of groups which are potential counterexamples to the Eilenberg–Ganea Conjecture, each of which has cohomological dimension 2. These are also examples of groups of type FP2 which are not finitely presented (see [1]). Dicks and Leary [3] give an explicit way of obtaining presentations (on finite generating sets) for these groups. For some of these examples, it is shown in [1] that any 2-dimensional classifying space would give rise to a counterexample to the Whitehead conjecture.We will refer to the examples cited above as Bestvina–Brady groups. These come equipped with natural, nonpositively curved cubical 3-dimensional classifying complexes, which we will call Bestvina–Brady complexes. In this short note, we show that these Bestvina–Brady complexes are (up to homotopy equivalence) formed by applying the Quillen plus construction to certain finite 2-complexes. From this, together with known facts about 2-complexes with aspherical plus constructions, we recover the result of Bestvina and Brady [1] that the Bestvina–Brady groups act freely on acyclic 2-complexes, and hence have cohomological dimension at most 2. It also follows that these groups have free relation modules of finite rank and so are of type FF. Finally, we use our construction to give an alternative proof of the cited theorem of Bestvina and Brady; at least one of the Eilenberg–Ganea and Whitehead conjectures is false.We do not use the full force of the Morse-theoretical techniques developed in [1], but will assume two results form that paper; the asphericity of the Bestvina–Brady complexes and the non-finite presentability of the Bestvina–Brady groups.

Homology of the universal covering of a co-H-space

The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-IS-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional. assumptions on the given co-H-structure. In this paper, we show a homological property of coH-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.

ON CATEGORY WEIGHT AND ITS APPLICATIONS

We develop and apply the concept of category weight which was introduced by Fadell and Husseini. For example, we prove that category weight of every Massey product 〈u 1, …, u n〉, u i∈ H ̃ ∗(X) is at least 2 provided X is connected. Furthermore, we remark that elements of maximal category weight enable us to control the Lusternik–Schnirelmann category of a space. For example, we prove that if f: N→M is a map of degree 1 of closed stable parallelizable manifolds and dim M⩽2 cat M−4 then cat N⩾ cat M . We also prove that if M is a closed manifold with dim M⩽2 cat M−3 then cat (M×S m 1 ×⋯×S m n )= cat M+n , i.e., the Ganea conjecture holds for M.

The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category

This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with Z/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space is completely determined. A second application shows that the Lusternik–Schnirelmann category for a large class of Seifert manifolds is equal to 3, which in turn is used to verify the Ganea conjecture for these Seifert manifolds.

On the Ganea conjecture for manifolds

On the Ganea conjecture for manifolds

Lower bounds for the L.-S. category of products

Halperin and Lemaire introduced L.-S. category type invariants left- Mcat ⁡ ( A ) {\text {left-}}\operatorname {Mcat} (A) and right- Mcat ⁡ ( A ) {\text {right-}}\operatorname {Mcat} (A) (/l) for certain differential algebras ( A , d ) (A,d) . In particular, they proved that if ( A , d ) = C ∗ ( S , k ) (A,d) = {C^{\ast }}(S,k) is the k k -valued singular cochains on 1 1 -connected space S S , then these invariants are lower bounds for the classical category cat ( S ) {\text {cat}}(S) . We use an explicit model for Ganea’s space due to Felix, Halperin, Lemaire, and Thomas to prove lMcat ⁡ ( A ⊗ B ) ⩽ lMcat ⁡ ( A ) + e ( B ) \operatorname {lMcat} (A \otimes B) \leqslant \operatorname {lMcat} (A) + e(B) , over any field, where e e denotes Toomer’s invariant. This proves Ganea’s conjecture for Mcat over fields of arbitrary characteristic.

A proof of ganea's conjecture for rational spaces

A proof of ganea's conjecture for rational spaces

Rational L-S category and conjecture of Ganea

The Lusternik-Schnirelmann category of a space satisfies cat( X× Y)≤cat( X)+cat( Y) and the inequality can be strict. However, Ganea conjectured that equality would hold if one factor was a sphere. Halperin and Lamaire recently introduced Mcat( X), a module-type approximation to cat( X) satisfying Mcat( X)≤cat( X). In some cases when F→ E→ B is a fibration, lower bounds for Mcat( E) are found in terms of Mcat( B) and invariants of F, and this proves Ganea's conjecture for Mcat.