Lusternik–Schnirelmann category and the connectivity ofX

Abstract

We define and study a homotopy invariant called the connectivity weight to compute the weighted length between spaces X and Y . This is an invariant based on the connectivity of A i , where A i is a space attached in a mapping cone sequence from X to Y . We use the Lusternik-Schnirelmann category to prove a theorem concerning the connectivity of all spaces attached in any decomposition from X to Y . This theorem is used to prove that for any positive rational number q , there is a space X such that q D cl ! .X /, the connectivity weighted cone-length of X . We compute cl ! .X / and kl ! .X / for many spaces and give several examples. 55M30, 55P05

On the same n-types for the wedges of the Eilenberg-Maclane spaces

“…As the dual notion of Hopf spaces with multiplications, co-H-spaces with comultiplications play a pivotal role in homotopy theory. We note that a non-contractible co-H-space is the space of Lusternik-Schnirelmann category 2 (see [28, Chapter X] and [27]); that is, cat(X) = 2 which is bigger than or equal to the R-cup length of the cohomology of this space with coefficients in a commutative ring R. One of the most important classes of co-H-spaces consists of all n-spheres for n ≥ 1, a wedge of spheres (or co-H-spaces), and the suspensions of a pointed space which we will deal with in this article.…”

On the generalized same N-type conjecture

“…If a comultiplication ϕ : X → X ∨ X is homotopy associative and has a right and left inverse, then for every space Z the set [X, Z ] of homotopy classes from X to Z becomes a group with the group operation depending on the comultiplication of a space. We observe that non-contractible co-H-spaces are precisely spaces of Lusternik-Schnirelmann category 2 (see [21, chapter X] and [20]). An important class of co-H-spaces consists of all n-spheres for n 1, a wedge of spheres, and the suspensions of a based space which we are dealing with in this paper.…”