What motivates the choice of chain complex when calculating simplicial homology? I've been confused by the idea of homology for a while but I've decided to approach it because of an interesting paper I read recently.
When I was first introduced to it the lecturer was talking about formal linear combinations of the simplices with coefficients from $\mathbb{Z}$. In the paper I read they computed homology using coefficients from $\mathbb{Z}_2$ What motivates the choice of a group when doing calculations?
 A: Choosing coefficients in $\Bbb{Z}_2$ allows one to disregard the orientation of simplices. This is used often in persistent homology to simplify computations. For instance, $C_n$ over $\Bbb{Z}_2$ would just be all possible combinations (the only non-trivial combinations are those with $1$ as a coefficient) of single $n$-dimensional simplices. It also follows that a chain’s inverse would be itself.
A: Look at the homology groups for non-orientable surfaces such as the Klein bottle $K$. Then $H_2(K,\Bbb Z)=0$ but $H_2(K,\Bbb Z_2)\cong\Bbb Z_2$.
So in this case homology over $\Bbb Z_2$ sees more than that over $\Bbb Z$.
Using $\Bbb Z_2$ coefficients is often useful when one has non-orientability.
A: A couple of reasons:


*

*Sometimes you don't have a choice of the group. De Rham cohomology, for example, is defined using differential equations on a smooth manifold and therefore naturally has coefficients in $\mathbb{R}$ or $\mathbb{C}$. (I've switched from talking about homology to cohomology, but they're closely related.)

*More complicated homology and cohomology theories can simliarly have natural coefficient groups, or even more complicated objects on which to act. The natural setting of group cohomology is a module on which the group acts, and more interesting modules than $\mathbb{Z}$ with the trivial action can often give you more interesting or more powerful results. Sheaf cohomology is more complicated than I can get into here, but one way of describing it (which should not be taken seriously) is that it's cohomology with a continually varying coefficient group.

*As was mentioned above, different homology and cohomology coefficients can allow you detect different stuff going on in the underlying topological space. 

*Poincare duality requires orientability. That means that it may not hold over $\mathbb{Z}$, but every manifold (or CW-complex, etc.) is orientable over $\mathbb{Z}_2$.

*More complicated homology and cohomology theories often involving counting something according to sign (say, induced from some sort of orientation), and switching from $\mathbb{Z}$ to $\mathbb{Z}_2$ eliminates the need to worry about it. (I think Floer homology falls into this category, but it's been a while.)


In practice, because of the universal coefficient theorem, the main groups people care about (for ordinary (co)homology, in the topological setting) are $\mathbb{Z}$ and $\mathbb{Z}_p$. Sometimes $\mathbb{Q}$ is useful to make a quick dimension argument rather than worrying about torsion.
