Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).


I need a reference to the following facts (which I believe are true at least in dimension $n=3$):


**Fact 1.** For every closed connected subset $A\subset X$ that can be embedded to $\mathbb R^{n-1}$ the complement $X\setminus A$ is connected.


**Fact 2.** For any closed subset $B\subset X$ whose connected components can be embedded to $\mathbb R$, the identity embedding $X\setminus B\to X$ induces an injective homomorphism $H\_1(X\setminus B;G)\to H\_1(X;G)$ in singular homologies with coefficients in some group $G$ (for example $\mathbb Z$ or $\mathbb Z/2\mathbb Z$).


**Remark.** The Alexander-Pontryagin Duality Theorem implies that Facts 1 and 2 are true if $X$ is the $n$-sphere. So, I need these facts for an arbitrary compact connnected $n$-manifold without boundary.



The homological negligibility of certain subsets in compact manifolds
Restating my comment above (which linked to another MO post):


For an open subspace $U\subset X$ there is a long exact sequence (via the normal LES for the pair $(X,X-U)$ and excision)
$$ \cdots\to H^\ast\_c(U) \to H^\ast\_c(X) \to H^\ast\_c(X-U) \to H^{\ast+1}\_c(U)\to\cdots$$
and Poincaré duality with compact supports $H^\ast\_c(M)\cong H\_{\dim M-\ast}(M)$, see for example Theorem 3.35 of Hatcher's bible.


Apply $\ast=n$ for Fact 1 (using $\dim A\le n-1$) and $\ast=n-1$ for Fact 2 (using $H^{n-2}(B)=0$).


