If C=C_ϕdenotes the mapping cone of an essential phantom map ϕfrom the suspension of the Eilenberg-Mac Lane complex K=K(Z,5) to the 4-sphere S=S^4 we derive the following properties: (1) The LS category of the product of C with any n-sphere S^n is equal to 3; (2) The LS category of the product of C with itself is equal to 3, hence is strictly less than twice the LS category of C. These properties came to light in the course of an unsuccessful attempt to find, for each positive integer m, an example of a pair of 1-connected CW-complexes of finite type in the same Mislin (localization) genus with LS categories m and 2m. If ϕis such that its p-localizations are inessential for all primes p, then by the main result of [J. Roitberg, The Lusternik-Schnirelmann category of certain infinite CW-complexes, Topology 39 (2000), 95-101], the pair C_*, C where C_*= S wedge Σ^2 K, provides such an example in the case m=1.
We show that the Lusternik-Schnirelmann category of the symplectic group Sp(3) is 5. This L-S category coincides with the cone length and the stable weak category.
In this paper we estimate the Lusternik-Schnirelmann category of the connected sum of two manifolds through their categories. We achieve a more general result regarding the category of a quotient space X/A where A is a suitable subspace of X.
In this paper, we give a new simplified calculation of the Lusternik-Schnirelmann category of closed 3-manifolds. We also describe when 3-manifolds have detecting elements and prove that 3-manifolds satisfy the equality of the Ganea conjecture.
It is a well known fact that formal dg-algebras admit no non-trivial Massey products, while the converse fails. We prove that by restricting to dg-algebras whose induced product on cohomology is trivial, we do in fact get this converse. This allows us to prove that spaces of Lusternik-Schnirelmann category 1 are formal spaces.
Given a map $f: M \to N$ of degree 1 of closed manifolds. Is it true that the Lusternik--Schnirelmann category of the range of the map is not more that the category of the domain? We discuss this and some related questions.
In this paper, we study the Lusternik-Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including its relevance to products and fibrations. We relate this category of a map to the classical Lusternik-Schnirelmann category of a map between finite topological spaces. Finally, we show how the simplicial category of a map may be used to define and study a simplicial version of the essential category weight of J. Strom.
This work was intended as an attempt to investigate a model of irreversible process and natural phenomena. For this, we introduce the notion of irreversible path (that for brevity we write ir-path), ir-homotopy, ir-contractible space, and Lusternik-Schnirelmann ir-category by equipping the $I=[0,1]$ with left order topology. We will restrict the irreversibility of definitions to $T_0$ Spaces, such that for $T_1$ spaces, the ir-paths are constant. After providing some theorems and properties of these notions, eventually, we prove that Lusternik-Schnirelmann ir-category is an invariant of ir-hom…
We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). We show that the non-existence of such critical objects guarantees the strong homotopy equivalence (in the Barmak and Minian's sense) between the corresponding sublevel complexes. Finally, we establish that the number of critical…
We are trying to look over the Lusternik-Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC.
The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound on the number of critical points of a smooth function on a manifold. Another similar invariant is the topological complexity $TC(X)$ (a la Farber) which has interesting applications in Robotics, specifically, in the robot motion planning problem. In this paper we calculate the Lusternik-Schnirelmann category and as a consequence we calculate the topological complexity of the two-point ordered configuration space of $\mathbb{CP}^n$ for every $n\geq 1$.
We prove that the LS category of the symplectic group $Sp(n)$ is bounded above by $\binom{n+1}{2}$. This is achieved by computing the number of critical levels of a height function.
We compute the Lusternik-Schnirelmann category and the topological complexity of no $k$-equal manifolds $M^{(k)}_d(n)$ for certain values of $d$, $k$ and $n$. This includes instances where $M^{(k)}_d(n)$ is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring $H^*(M^{(k)}_d(n))$ as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.
The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms, that generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that allows to develop all the machinery of algebraic topology which is costumary in the classical theory of Lusternik-Schnirelmann category.
We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given.
In this article we study a homotopy invariant cat(X,B,ξ) on a pair of finite CW complexes with respect to a continuous closed 1-form. This is a generalisation of a Lusternik-Schnirelmann category developed by Farber, studying the topology of a closed 1-form. The article establishes the connection with the original notion and obtains analogous results on critical points and homoclinic cycles. We also provide a similar cuplength lower bound for cat(X,B,ξ).
In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between sim…
A criterion to determine the L-S category of a total space of a sphere-bundle over a sphere is given in terms of 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 such a total space $N$ to have the same L-S category as its `once punctured submanifold' $N\smallsetminus\{P\}$, $P \in N$. Also a necessary condition for such a total space $M$ to satisfy Ganea's conjecture is obtained.
We determine the Lusternik-Schnirelmann category of the irreducible, symmetric Riemann spaces SU(n)/SO(n) and SU(2n)/Sp(n) of type AI and AII respectively.
We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids.
We use the Berstein-Hilton invariant to prove the formula $\cat(M_1\sharp M_2)=\max\{\cat M_1, \cat M_2\}$ for the Lustrnik-Schnirelmann category of the connected sum of closed manifolds $M_1$ and $M_2$.
We develop Morse-Bott theory on posets, generalizing both discrete Morse-Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik-Schnirelmann theorem for general matchings on posets, in particular, for Morse-Bott functions.
We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.
We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold M, the dimension of M, and the Lusternik-Schnirelmann category of M, and relate the latter to the systolic category of M.
Farber and Rudyak introduced topological complexity $\mathbf{TC}(X)$ of motion planning and its higher analogs $\mathbf{TC}_n(X)$ to measure the complexity of assigning paths to point tuples. Motivated by motion planning where a robotic system starts at the home configuration and possibly comes back after passing through a list of locations, we define three other classes of topological complexities $\mathbf{LTC}_n(X)$, $\mathbf{ltc}_n(X)$ and $\mathbf{tc}_n(X)$. We will compare these notions and compute the latter for some familiar classes of spaces.
We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds.
We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lower bound of 3 for systolic category.
We extend the theory of the Lusternik-Schnirelmann category to general metric spaces by means of covers by arbitrary subsets. We also generalize the definition of the strict category weight. We show that if the Bockstein homomorphism on a metric space is non-zero, then its LS-category is at least two, and use this to compute the category of Pontryagin surfaces. Additionally, we prove that a Polish space with LS-category $n$ can be presented as the inverse limit of ANR spaces of category at most $n$.
We introduce the notion of inductive category in a model category and prove that it agrees with the Ganea approach given by Doeraene. This notion also coincides with the topological one when we consider the category of (well-) pointed topological spaces.
To clarify the method behind the paper "Ganea's conjecture on Lusternik-Schnirelman category" by the author, 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 "Ganea's ..." for all the cases except the case when …
Let $F \hookrightarrow X \to B$ be a fibre bundle with structure group $G$, where $B$ is $(d{-}1)$-connected and of finite dimension, $d \geq 1$. We prove that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim B}{d}$, 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 $\cat{PU(n)} \leq 3(n{-}1)$ for all $n \geq 1$, which might be best possible, since we have $\cat{\mathrm{PU}(p^r)}=3(p^r{-}1)$ f…
Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(π)$ denotes the cohomological dimension of a group $π$ and ${\rm hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: \[ {\rm cat} X\leq \Bigl\lceil\frac{{\rm hd} (X)}{k+1}\Bigr\rceil \text{ if } π_i(X)=0 \text{ for } i\leq k. \] We make a synthesis and generalization of both of these results, by demonstrating the main result: \[ {\rm cat}\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{k+1}\…
Let $G$ be a compact connected Lie group and $p : E\to ΣA$ be a principal G-bundle with a characteristic map $α: A\to G$, where $A=ΣA_{0}$ for some $A_{0}$. Let $\{K_{i}{\to} F_{i-1}{\hookrightarrow} F_{i} \,|\, 1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} Σ{K_{1}} \; \text{and}\; F_{n}{\simeq} G \}$ 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 $α$…
Let G be a compact connected Lie group and p : E \to Σ^2V a principal G-bundle with a characteristic map α: A=ΣV \to G. By combining cone decomposition arguments in Iwase-Mimura-Nishimoto [3,5] with computations of higher Hopf invariants introduced in Iwase [8], we generalize the result in Iwase-Mimura [12]: Let {F_{i}|0 \leq i \leq m} be a cone-decomposition of G with a canonical structure map σ_{i} of cat(F_{i}) \leq i for i \leq m. We have cat(E) \leq \Max(m+n,m+2) for n \geq 1, if αis compressible into F_{n} \subseteq F_{m} \simeq G and H^{σ_n}_n(α) = 0, under a suitable compatibility cond…
In [3], after defining notions of LS category in the simplicial context, the authors show that the geometric simplicial LS category is non-decreasing under strong collapses. However, they do not give examples where it increases strictly, but they conjecture that such an example should exist, and thus that the geometric simplicial LS category is not strong homotopy invariant. The purpose of this note is to provide with such an example. We construct a simplicial complex whose simplicial and geometric simplicial LS categories are different, and using this, we provide an example of a strong collap…
We prove the discrete Lusternik-Schnirelmann theorem telling that tcat(G) less or equal to crit(G) for a general simple graph G=(V,E). It relates the minimal number tcat(G) of in G contractible graphs covering G, with crit(G), the minimal number of critical points which an injective function f on the vertex set V can have. We also prove that the cup length cup(G) is less or equal to tcat(G) which is valid also for any finite simple graph. If cat(G) is the minimal tcat(H) among all graphs H homotopic to G and cri(G) is the minimal crit(H) among all graphs H homotopic to G, we get a relation bet…
We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and defined in terms of the existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns …
The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik-Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times\mathbb{R}^n$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.
Extending a result of Félix-Halperin-Lemaire on Lusternik-Schnirelmann category of products, we prove additivity of a rational approximation for Schwarz's sectional category with respect to products of fibrations.
In this paper some axiomatic generalization (function of open subsets) of the relative Lyusternik-Schnirelmann category is considered, incorporating the sectional category and the Schwarz genus as well. For this function and a given continuous map of the underlying space to a finite-dimensional metric space some lower bounds on the value of this function on the (neighborhood of) preimage of some point are given.
We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtained an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.
We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.
By analogy with the invariant Q-category defined by Scheerer, Stanley and Tanré, we introduce the notions of Q-sectional category and Q-topological complexity. We establish several properties of these invariants. We also obtain a formula for the behaviour of the sectional category with respect to a fibration which generalizes the classical formulas for Lusternik-Schnirelmann category and topological complexity.
The present paper is a continuation of author's paper arXiv:1909.00940 [math.AT] devoted to the lemmas of Alexander and Sperner, but is independent from it. We begin by a step back from Alexander and Sperner to Lebesgue work on the invariance of the dimension. In contrast with almost everybody else, Lebesgue worked with cubes rather than with simplices. His methods were developed by Hurewicz and Lusternik-Schnirelmann and then forgotten. In the present paper these methods are recast in the language of cubical chains and cochains. After this, we present a new approach to Lebesgue and Lusternik-…
In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously known `topological complexity' of a map. Moreover, we study the relation between Lusternik-Schnirelmann category and topological complexity of two closed oriented manifolds connected by a degree one map.
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$. For topological complexity we prove the inequality $\TC (M\# N)\ge\max\{\TC M,\TC N\}$ for simply connected manifolds.
Let $X$ be a two-cell complex with attaching map $α\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1<q\leq3(p-1)$ under fairly mild conditions by making use of the Hopf invariant techniques recently developed by the authors in their study of the sectional category of arbitrar…
West Chester University（西切斯特大学）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）
