# Thread: does pushforward and the operator partial communicates with each other?

1. ## does pushforward and the operator partial communicates with each other?

we know that pushforward and d is commutative.
i'm confused that whether pushforward coomunicate with partial and conjugation?
what's more,i wonder whether it's true that pushforward composited with pullback is identity.
thx

2. ## Re: does pushforward and the operator partial communicates with each other?

The pullback of the pushforward is not necessarily the identity--consider the case where the codomain has lower dimension than the domain, for example. If your map is a local embedding, however, the result will be true.