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?



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 $Y$.

  • $\begingroup$ 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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