Abstract:
Let $X$ be a simply connected CW-complex of dimension $n$. This paper aims to establish a link between the groups $\mathcal{E}(X)$ and $\Gamma_n(X)$. Here, $\mathcal{E}(X)$ represents the group of self-homotopy equivalences of $X$, while the group $\Gamma_n(X)$, which was intruduced by Whitehead, is the image of $\pi_n$ of the $(n-1)$-skeleton of $X$ in $\pi_n$ of the $n$-skeleton of $X$. This pursuit provides significant insights into discerning the presence of torsion elements within $\mathcal{E}(X)$.
Key words and phrases:whitehead's exact sequence, group of self-homotopy equivalences.
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