The exterior product $\wedge$ is inherited in cohomology:

For $\wedge$ to be well-defined, we have to check two properties:

1. If $\alpha$ and $\beta$ are closed, $\alpha\wedge\beta$ is also closed:

Therefore,

induces

and also

2. If moreover $\alpha$ or $\beta$ is exact, $\alpha\wedge\beta$ is also exact:

Therefore,

induces