TITLE: $\mathbb{Z}/2\mathbb{Z}$ coefficients in homology
QUESTION [10 upvotes]: I don't see the point in using homology and cohomology with coefficients in the field $\mathbb{Z}/2\mathbb{Z}$.
Can you provide some examples for why this is useful?

REPLY [12 votes]: A lot of times it's easier to work with $\mathbb Z_2$ coefficients. You don't have to worry about sign and calculations are easier. 
Field coefficients are nice because there is a precise duality between homology and cohomology, without worrying about the universal coefficient theorem.
Knowledge of $\mathbb Z_2$ coefficients can help you say things about integral homology. For example, if $H_i(X;\mathbb Z_p)=0$ for all primes $p$, then $H_i(X;\mathbb Z)=0$. 
Some spaces have a nicer presentation with $\mathbb Z_2$ coefficients. The cohomology ring $H^*(\mathbb{RP}^n;\mathbb Z_2)$ is isomorphic to $\mathbb Z_2[x]/x^{n+1}$, whereas the integral cohomology ring is messier to write down.