We first compute the James' sectional category (secat) of the Ganea map g_k of any map f in terms of the sectional category of f: We show that secat(g_k) is the integer part of secat(f)/(k+1). Next we compute the relative category (relcat) of g_k. In order to do this, we introduce the relative category of order k (relcat_k) of a map and show that relcat(g_k) is the integer part of relcat_k(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcat_k(f) <= secat(f) + k + 1 and relcat_k(f) <= relcat_(k+1)(f) <= relcat_k(f) + 1. We give examples that show that these inequalities may be strict.
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.
We introduce the notion of fibrewise sectional category via a Whitehead-Ganea construction. Fibrewise sectional category is the analogue of the ordinary sectional category in the fibrewise setting and also the natural generalization of the fibrewise unpointed LS category in the sense of Iwase-Sakai. On the other hand the fibrewise pointed version is the generalization of the fibrewise pointed LS category in the sense of James-Morris. After giving their main properties we also establish some comparisons between such two versions.
We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied in the study of Farber's topological complexity for 2-cell complexes, as well as in the construction of a counter-example to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category.
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.
We prove that the sectional category of the universal fibration with fibre X, for X any space that satisfies a well-known conjecture of Halperin, equals one after rationalization.
We give simple upper bounds for rational sectional category and use them to compute invariants of the type of Farber's topological complexity of rational spaces. In particular we show that the sectional category of formal morphisms reaches its cohomological lower bound and give a method to compute higher topological complexity of formal spaces in terms of their cohomology.
Félix, Halperin, and Lemaire have shown that the rational module category Mcat and the rational Toomer invariant $e_0$ coincide for simply connected Poincaré duality complexes. We establish an analogue of this result for the sectional category of a fibration.
We give a simple algebraic characterisation of the sectional category of rational maps admitting a homotopy retraction. As a particular case we get the Félix-Halperin theorem for rational Lusternik-Schnirelmann category and prove the conjecture of Jessup-Murillo-Parent on rational topological complexity. We also give a characterisation for relative category in the sense of Doeraene-El Haouari.
We study the behavior of the abstract sectional category in the Quillen, the Strom and the Mixed proper model structures on topological spaces and prove that, under certain reasonable conditions, all of them coincide with the classical notion. As a result, the same conclusions hold for the abstract Lusternik-Schnirelmann category and the abstract topological complexity of a space.
This survey is a guide for the non specialist on how to use rational homotopy theory techniques to get approximations of Farber's topological complexity, in particular, and of Schwarz's sectional category, in general.
We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.
For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $π_{2,1}^X:F(X,2)\to X$, and the fixed point property (FPP) for self-maps on $X$. Explicitly, we demonstrate that a space $X$ has the FPP if and only if 2 is the minimal cardinality of open covers $\{U_i\}$ of $X$ such that each $U_i$ admits a continuous local section for $π_{2,1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.
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.
On the $m$-dimensional sectional category and induced invariants
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；Indian Statistical Institute（印度统计研究所）
If a map $f$ has a homotopy retraction, then Doeraene and El Haouari conjectured that the sectional category and the relative category of $f$ are the same. In this work we discuss this conjecture for some lower bounds of these invariants. In particular, when we consider the diagonal map, we obtain results supporting Iwase-Sakai's conjecture which asserts that the topological complexity is the monoidal topological complexity.
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…
Rudyak's conjecture states that cat$(M) \geq$ cat$(N)$ given a degree one map $f:M \to N$ between closed manifolds. We generalize this conjecture to sectional category, and follow the methodology of [5] to get the following result: Given a normal map of degree one $f:M \to N$ between smooth closed manifolds, fibrations $p^M:E^M \to M$ and $p^N:E^N \to N$, and lift $\overline{f}$ of $f$ with respect to $p^M$ and $p^N$, i.e., $fp^M = \overline{f} p^N$; then if $f$ has no surgery obstructions and $N$ satisfies the inequality $5 \leq \dim N \leq 2r$ secat$(p^N) - 3$ (where the fiber of $p^N$ is $(…
