Let $M$ be a manifold space, Let $H^1(M,\mathbb R)=\{0\}$, then is it possible to get a submanifold $S$ of $M$. such that $H^1(S, \mathbb R)\neq \{0\}$.
If $M$ is simply connected then we can remove a suitable point and then remaining open subset will be sub manifold and which will have abelian fundamental group and above statement is valid. But for general manifold where $H^1(M)=0$ is not because of simply conectedness, Can we get sub manifold?