We raise the problem of realisability of rings as $\{X,X\}$ the ring of stable homotopy classes of self-maps of a space $X$. By focusing on $A_n^2$-polyhedra, we show that the direct sum of three endomorphism rings of finitely-generated abelian groups, one of which must be free, is realisable as $\{X,X\}$ modulo the acyclic maps. We also show that $\mathbb{Z}_p^3$ is not so realisable, for $p$ any prime.

中文翻译：

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An2-多面体自映射的稳定同伦类环

我们将环的可实现性问题提出为 $\{X,X\}$ 空间 $X$ 的自映射的稳定同伦类环。通过关注 $A_n^2$-多面体，我们证明了有限生成阿贝尔群的三个自同态环的直接和，其中一个必须是自由的，可以实现为 $\{X,X\}$ 模非循环地图。我们还表明 $\mathbb{Z}_p^3$ 不是那么可实现的，因为 $p$ 是任何素数。