Given a group $G$, we provide a constructive method to get infinitely many (non-homotopy-equivalent) Alexandroff spaces, such that the group of autohomeomorphisms, the group of homotopy classes of self-homotopy equivalences and the pointed version are isomorphic to $G$. As a result, any group $G$ can be realized as the group of homotopy classes of self-homotopy equivalences of a topological space $X$, for which there exists a CW complex $\mathcal{K}(X)$ and a weak homotopy equivalence from $\mathcal{K}(X)$ to $X$.

中文翻译：

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Alexandroff 空间中群的拓扑实现

给定一个群$G$，我们提供了一个构造方法来获得无限多个（非同伦等价的）Alexandroff 空间，使得自同胚群、自同伦等价的同伦类群和尖点版本同构为$G$。因此，任何群$G$都可以实现为拓扑空间$X$的自同伦等价的同伦类群，其中存在一个CW复数$\mathcal{K}(X)$和一个从 $\mathcal{K}(X)$ 到 $X$ 的弱同伦等价。