Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by D, over the CW complex. The definition of “diagram of spaces” goes as follows: The following data constitute what is called a diagram of topological spaces D over a cell complex A: 1)for each vertex v of A we have a topological space D(v); 2)for each edge (v-w) we have a continuous map D(v-w): D(v)-D(w). Additionally, we require that these maps commute over each triangle in A. Given this diagram, we can take the colimit to get the “total space”. Also, a manifold can be considered as a colimit of its atlas. Or equivalently, an atlas is essentially a way of viewing a manifold as a colimit of Euclidean balls. It seems to me that the constructions of manifold and “colimit of diagram of spaces” have some similarities. So my first question is, can a “diagram of spaces over a CW complex” be a manifold? If so, under what conditions? In addition, I also read about the notion of “homotopy colimit”. According to the “Homotopy Lemma”, when all the spaces D(v) in the diagram are contractible, the homotopy colimit of the diagram is homotopy equivalent to the base complex. Since any Euclidean space is contractible, if all the spaces D(v) in the diagram are Euclidean, can I construct a manifold with “homotopy colimit”? (If yes, the manifold should be homotopy equivalent to the base CW complex by the “homotopy lemma”)