Homology, Homotopy and Applications

Volume 21 (2019)

Number 2

Spaces with complexity one

Pages: 23 – 26

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n2.a2

Author

Alyson Bittner (Department of Mathematics, University at Buffalo, State University of New York, Buffalo, N.Y., U.S.A.)

Abstract

An $A$-cellular space is a space built from a space A and its suspensions, analogous to the way that $CW$-complexes are built from $S^0$ and its suspensions. The $A$-cellular approximation of a space $X$ is an $A$-cellular space $C W_A X$, which is closest to $X$ among all $A$-cellular spaces. The $A$-complexity of a space $X$ is an ordinal number that quantifies how difficult it is to build an $A$-cellular approximation of $X$. In this paper, we study spaces with low complexity. In particular, we show that if $A$ is a sphere localized at a set of primes then the $A$-complexity of each space $X$ is at most $1$.

Keywords

cellular space, complexity, mapping space

2010 Mathematics Subject Classification

18C10, 55P60

Received 5 February 2018

Received revised 6 September 2018

Published 19 December 2018