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I am aware of tools for computing reduced homology when dealing with nonempty simplicial complexes, but:

What would be an effective approach for computing the reduced homology of the empty simplicial complex?

Any constructive responses would be greatly appreciated.

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Have you actually tried to do this? What approaches have you tried? Do you remember the definitions for each of the things you're trying to do? –  KReiser Mar 11 '12 at 21:24
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up vote 3 down vote accepted

You don't need any "tools" to compute the homology (singular, simplicial or otherwise) of an empty simplicial complex. Because the free abelian group on an empty set of generators is the trivial group, the chain groups are trivial, so $H_*(\emptyset) = 0$ (where by a slight abuse of notation $0$ denotes the group or graded group comprising just the identity element $0$).

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The OP was asking about the reduced homology. This is nontrivial in dimension $-1$ where $\tilde{H}_{-1}(\emptyset)=\mathbb{Z}$. This follows since $\tilde{C}_n=0$ for all $n\neq -1$ while $\tilde{C}_{-1}=\mathbb{Z}$. –  Adam Mar 12 '12 at 3:53
    
Agreed. I missed the word "reduced" in the question. –  Rob Arthan Mar 16 '12 at 16:58
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