Algebraic & Geometric Topology

Lusternik–Schnirelmann category and the connectivity of $X$

Nicholas A Scoville

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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

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

Received: 25 August 2011
Revised: 8 December 2011
Accepted: 8 December 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 55P05: Homotopy extension properties, cofibrations

Lusternik–Schnirelmann category categorical sequence cone length killing length Egyptian fractions mapping cone sequence


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.

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