This questions is related to oxygen sensors on cars-trucks- Does an oxygen sensor detect lack of oxygen underneath the carriage of a vehicle? Is it possible that an oxygen sensor will cause the engine light to turn on because it is working due to the manifold exhaust being cracked? Thanks

What happens is the ecm looks at voltages of the o2 sensors. Usually between 0-1V. If it hovers around a half a volt, it's considered happy because it's seeing the o2 from the exhaust it wants to see.
Now open a crack in the exhaust manifold like you describe, and you are entering more oxygen because it's sucking it in. So, down stream, the o2 sensor sees this and ups the voltage to make the injectors run richer to make up for it. Sometimes it's too much then triggers the CEL on.
So to answer your question... Yes it does!