# Help with gluing together surfaces of infinite genus

Let $$S=\bigwedge_{m\in\mathbb{Z}}\mathbb{S}^{1}$$ denote the wedge sum of countably infinitely many circles—that is, the topological space obtained by removing countably infinitely many distinct points from, say, $$\mathbb{C}$$: $$S=\mathbb{C}\backslash\left\{ z_{m}:m\in\mathbb{Z}\right\}$$. Let $$\nu$$ be an integer $$\geq2$$, and suppose we have $$\nu$$ copies of $$S$$ (written $$S_{1},\ldots,S_{\nu}$$) to be glued together along the “holes” (ex: for each $$m$$, the surfaces $$S_{1},\ldots,S_{\nu}$$ get glued together at the hole at $$z_{m}$$). I'm pretty sure that the resultant topological space (call it $$T$$) is then: $$T=\prod_{n=1}^{\nu}S=\prod_{n=1}^{\nu}\bigwedge_{m\in\mathbb{Z}}\mathbb{S}^{1}$$ that is, $$T$$ is the direct product of $$\nu$$ copies of $$S$$. Is there a simpler description of $$T$$ (i.e., is there a way to interchange $$\prod$$ and $$\bigwedge$$; is there a way to write things in terms of $$\mathbb{T}^{k}$$ for some $$k$$s (where $$\mathbb{T}^{k}=\prod_{i=1}^{k}\mathbb{S}^{1}$$), etc.)?

• Be careful. The wedge sum of circles is not the same topological space as $\mathbb{C}$ without some points. They are homotopy equivalent, but not topologically the same. It is not clear to me how you would glue something together along a hole. Perhaps along the boundary of a hole? – Strichcoder Mar 25 at 22:15