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I have a pair of complexes of modules $A_{\bullet}$ and $B_{\bullet}$, and I want to create a new complex pretty trivially by shifting one complex by 1, and then taking the direct sum of the complexes. That is, I want $C_{\bullet}$ such that $C_n=A_{n-1}\oplus B_n$. The differential will simply be componentwise using the original maps. As it turns out, it is pretty easy to represent this new complex as a mapping cone if I let $f:A_{\bullet}\rightarrow B_{\bullet}$ be the zero map on each module. That is, $C_{\bullet}=\text{Cone}(f)$.

My question comes in the topological interpretation of what I am doing here. I am considering these complexes algebraically, but the modules are generated by a labeled simplicial complex, so there is some topology around. So, to be more concise, what is the topological interpretation of the zero map between two simplicial complexes?

Thanks.

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It's the constant map. The mapping cone of a constant map $X \to Y$ is just the wedge of the suspension $\Sigma X$ and $Y$.

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  • $\begingroup$ This is exactly what I was looking for. I knew it had to have a simple interpretation that I just didn't know. Thanks. $\endgroup$ – Trevor Oct 28 '13 at 14:05

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