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I am interested in showing that the dimensions of certain homology groups for some abstract simplicial complexes are positive. I would like to know if there is any procedure of showing this inductively by splitting vertices of the simplicial complex and obtaining a larger complex each time. For example, start with an empty tetrahedron on the vertices $\{1,2,3,4\}$ (so the facets of the complex are $4$ filled triangles) and split $1$ to obtain a complex with the vertex set $\{1,2,3,4,5\}$ in which some of the old facets may become unavailable, but new ones might appear. I should also mention that these complexes are not necessarily pure. Could you please point out any tools that might be helpful? Could simplical maps help or are there ways to extend a cycle from the smaller complex to the larger one? Also, if you can think of some general methods to estimate the dimensions of the homology groups of a complex, I would be interested.
Thank you.

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Why do you want to add more vertices/edges/faces/... to your simplicial complex? When you talk about finite simplicial complexes, you can calculate their homology by hand, and the smaller the data, the smaller the calculations (I would reckon), especially if the simplicial complex you begin with is as well behaved as the shell of a tetrahedron. What is your motivation for this? –  Olivier Bégassat Jul 21 '12 at 21:27
    
I do not think your idea of doing triangulation would be helpful in this case. Also, by definition we should have $G\cong \mathbb{Z}^{n}\oplus F$ where $F$ is a torsion group.I do not see how the dimension could be negative. The homology group could be trivial, but computing $H_{1}(X)$ usually is a simply matter as you only need the vertices and edges. –  Bombyx mori Jul 22 '12 at 0:53
    
I am working with arbitrarily large simplicial complexes and I have a method of constructing them inductively starting from something simple like an empty tetrahedron. I should also mention that I need to compute the dimensions of the homology groups over $\mathbb{C}$. When I said I want positive dimensions I wanted to imply that I need them to be nonzero. I realize that my question is not very well expressed, but to present it rigorously would require me to introduce a lot of tedious notation. Thank you. –  user36260 Jul 23 '12 at 3:03
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