Journal of the
Korean Mathematical Society JKMS

Given a topological space $X$ and a non-negative integer $k$, $\E_{k}^{\sharp}(X)$ is the set of all self-homotopy equivalences of $X$ that do not change maps from $X$ to an $t$-sphere $S^{t}$ homotopically by the composition for all $t \geq k$. This set is a subgroup of the self-homotopy equivalence group $\E(X)$. We find certain homotopic tools for computations of $\E_{k}^{\sharp}(X)$. Using these results, we determine $\E_{k}^{\sharp}(M(G,n))$ for $k\geq n$, where $M(G, n)$ is a Moore space type of $(G, n)$ for a finitely generated abelian group $G$.

Keywords: self-homotopy equivalence, cohomotopy group, Moore space, co-Moore space

MSC numbers: Primary 55P10, 55Q05, 55Q55

Supported by: The first-named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of education(NRF-2015R1C1A1A01055455).
The second-named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2018R1D1A1B07045599).