What is a relevant non commutative analogues for the following fact, in term of spectral triples and cyclic cohomology?:
"If $M$ is a compact oriantable manifold without boundary and $X\subset M$ is a proper subset with inclusion $i: X \to M$, then $i^*$ is not a ring isomorphism in cohomology"
I asked the commutative version in the following post:
A closed manifold with a subset with the same ring cohomology
