In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a \(\Delta \)-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a large family of both \(\Delta \)-complexes, as well as simplicial complexes, which are contractible, but not collapsible.

中文翻译：

---

炒简单的拓扑

在本文中，我们定义了一个拓扑空间族，其中包含并广义地概括了较高维的Dunce帽子。我们的定义纯粹是组合的，用标准d -simplex的边界单纯形的标识来表述。通过这种构造，获得的空间可以用单词索引，并且它们自动携带 \（\ Delta \）的结构-复杂。作为我们的主要结果，我们完全确定了这些空间的同伦类型。实际上，有些令人惊讶的是，我们能够证明它们中的每一个都是可收缩的或同质的，等效于奇维球体。我们开发了直接从索引词的组合来确定同型类型的语言。作为我们研究的附加好处，我们能够模仿Dunce hat现象，并获得可压缩但不可折叠的\（\ Delta \）复合物以及简单复合物的大家族。