#### Volume 12, issue 1 (2012)

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Lusternik–Schnirelmann category and the connectivity of $X$

### Nicholas A Scoville

Algebraic & Geometric Topology 12 (2012) 435–448
##### 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.

##### Keywords
Lusternik–Schnirelmann category, categorical sequence, cone length, killing length, Egyptian fractions, mapping cone sequence
##### Mathematical Subject Classification 2010
Primary: 55M30, 55P05