Certain self-homotopy equivalences on wedge products of Moore spaces

Ho Won Choi and Kee Young Lee

Vol. 272 (2014), No. 1, 35–57

DOI: 10.2140/pjm.2014.272.35

Abstract

For a based 1-connected finite CW-complex $X$, let $\mathcal{ℰ}\left(X\right)$ denote the group of homotopy classes of self-homotopy equivalences on $X$, and ${\mathcal{ℰ}}_{♯}^{dim+r}\left(X\right)$ the subgroup of $\mathcal{ℰ}\left(X\right)$ of homotopy classes of self-homotopy equivalences on $X$ that induce the identity homomorphism on the homotopy groups of $X$ in dimensions $\le dimX+r$. For two given Moore spaces $M_1=M\left(Z_q,n+1\right)$ and $M_2=M\left(Z_p,n\right)$ with $n\ge 5$, we investigate the subsets of $\left[M_1,M_2\right]$ and $\left[M_2,M_1\right]$ consisting of homotopy classes of maps that induce the trivial homomorphism between the homotopy groups of $M_1$ and those of $M_2$ in dimensions $\le dimX+r$. Using the results of this investigation, we completely determine the subgroups ${\mathcal{ℰ}}_{♯}^{dim+r}\left(M\left(Z_q,n+1\right)\vee M\left(Z_p,n\right)\right)$, where $p$ and $q$ are positive integers, for $n\ge 5$ and $r=0,1$.

Keywords

self-homotopy equivalence, Moore space, homotopy group

Mathematical Subject Classification 2010

Primary: 55P10, 55Q05, 55Q20

Secondary: 55Q40, 55Q52

Milestones

Received: 2 August 2013

Revised: 17 October 2013

Accepted: 23 October 2013

Published: 9 October 2014