What portion of the brain connects the two cerebral hemispheres?

I think what you are looking for is the corpus callossum. The corpus callossum connects the two cerebral hemispheres. It acts a lot like a bridge, that allows the right and left hemispheres to communicate with one another. Without this, each of our eyes would only send information to one hemisphere, instead of to both sides of the brain. This is evident from studies done on patients who have undergone split-brain surgery. Check the YouTube video in the related links out for an example of what I mean, it's quite interesting!: