Lusternik–Schnirelmann category and the connectivity of X

Scoville, Nicholas A

doi:10.2140/agt.2012.12.435

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Lusternik–Schnirelmann category and the connectivity of $X$

Nicholas A Scoville

Algebraic & Geometric Topology 12 (2012) 435–448

DOI: 10.2140/agt.2012.12.435

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 $q = {cl}^{ω} (X)$ , the connectivity weighted cone-length of $X$ . We compute ${cl}^{ω} (X)$ and ${kl}^{ω} (X)$ 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

References

Publication

Received: 25 August 2011

Revised: 8 December 2011

Accepted: 8 December 2011

Published: 20 March 2012

Authors

Nicholas A Scoville
Mathematics and Computer Science Ursinus College 610 E Main Street Collegeville PA 19426 USA
http://webpages.ursinus.edu/nscoville/

Volume 12, issue 1 (2012)