$\mathbb{Z}/2\mathbb{Z}$ coefficients in homology

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?

Solution 1:

1. 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.

2. Field coefficients are nice because there is a precise duality between homology and cohomology, without worrying about the universal coefficient theorem.

3. 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 $H_i(X;\mathbb Z)=0$.

4. 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.

Solution 2:

Here is another one to add to the list: I am currently reading about cohomology operations, in particular the Steenrod squares.

These are functions

$$\operatorname{Sq^i}:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$$

which satisfies a whole bunch of nice properties (or axioms depending on how one looks at things)

Adams used these cohomology operations to solve the vector fields on spheres problem.

See the very excellent book by Mosher & Tangora's "Cohomology Operations and Applications in Hommotopy Theory" for the details