Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

The categorical sequence of a rational space

Pages: 49 – 71

DOI: http://dx.doi.org/10.4310/HHA.2019.v21.n1.a3

Authors

Julienne Dare Houck

Jeffrey Strom (Department of Mathematics, Western Michigan University, Kalamazoo, Mi., U.S.A.)

Abstract

The categorical sequence of a space $X$ is a sequence of integers that encodes the growth of the Lusternik–Schnirelmann category of its $\textrm{CW}$ skeleta as dimension increases. Restrictions on these sequences found in “Categorical sequences” [R. Nendorf, N. Scoville, and J. Strom. Algebr. Geom. Topol., 6:809–838, 2006] have proven to be powerful tools in studying and computing $\textrm{L-S}$ category, motivating the search for additional restrictions. In this paper we study the initial three-term segments of the categorical sequences of rational spaces of finite type.We show that there is another restriction: a sequence of the form $(a, b, a + b, \dotsc)$ is the categorical sequence of a rational space of finite type if and only if $b \equiv 2 \: \mathrm{mod} \: a - 1$. With the possible exception of a small number of values of $c$ for each $a$, all other three-term initial sequences are realizable by simply-connected rational spaces of finite type.

Keywords

Lusternik–Schnirelmann category, rational homotopy theory

2010 Mathematics Subject Classification

55M30, 55P62, 55Q15

Full Text (PDF format)

Received 12 January 2018

Published 22 August 2018