Lusternik–Schnirelmann category and the connectivity of $X$

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 $q={cl}^{\omega }\left(X\right)$, the connectivity weighted cone-length of $X$. We compute ${cl}^{\omega }\left(X\right)$ and ${kl}^{\omega }\left(X\right)$ for many spaces and give several examples.