Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{\*}$ between $\mathbb{Z}$-cohomologies?


In this question $A$ is merely a proper subset.(not necessarily compact, not necessarily submanifold)



A closed manifold with a subset with the same ring cohomology
Suppose for simplicity that $M$ is connected and orientable. Choose a point $x\in M\setminus A$, and a closed disc $U$ centred at $x$. We can use the chart to deform $i$ into a homotopic map $j$ such that $j(A)\subseteq M\setminus\text{int}(U)$. Collapsing the complement of $U$ gives a map $p$ from $M$ to the one-point compactification $U\cup\{\infty\}$, which is homeomorphic to $S^n$. It is standard that the resulting map $p^\*\colon\mathbb{Z}=\widetilde{H}^n(S^n)\to H^n(M)$ is an isomorphism, but $pj$ is constant, so the map $i^\*p^\*=j^\*p^\*=(pj)^\*$ is zero, so $i^\*$ is not an isomorphism.


