## Algebraic & Geometric Topology

### Lusternik–Schnirelmann category and the connectivity of $X$

Nicholas A Scoville

#### 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 $Ai$, where $Ai$ 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 $q= clω(X)$, the connectivity weighted cone-length of $X$. We compute $clω(X)$ and $klω(X)$ for many spaces and give several examples.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 1 (2012), 435-448.

Dates
Revised: 8 December 2011
Accepted: 8 December 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715345

Digital Object Identifier
doi:10.2140/agt.2012.12.435

Mathematical Reviews number (MathSciNet)
MR2916282

Zentralblatt MATH identifier
1250.55001

#### Citation

Scoville, Nicholas A. Lusternik–Schnirelmann category and the connectivity of $X$. Algebr. Geom. Topol. 12 (2012), no. 1, 435--448. doi:10.2140/agt.2012.12.435. https://projecteuclid.org/euclid.agt/1513715345

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