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# Homology, Homotopy and Applications

## Volume 21 (2019)

### Number 1

### The categorical sequence of a rational space

Pages: 49 – 71

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

#### Authors

#### 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

Received 12 January 2018

Published 22 August 2018