If the head of cockroach is removed, it may live for few days because :

The head holds a bit of a nervous system while the rest is situated along the ventral (belly-side) part of its body. So, if the head of a cockroach is cut off, it will still live for as long as one week. In the head region, the brain is represented by supra-oesophageal ganglion which supplies nerves to antennae and compound eyes.