Let $M$ and $N$ be two smooth compact, oriented manifolds and $X\subset M$ an oriented submanifold of $M$ of dimension $k$ (not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained inside a submanifold of dimension $k-2$ or less, where $\bar{X}$ denotes the closure of $X$ inside $M$. Let $f:M \rightarrow N$ be a smooth map such that $Y := f(X) \subset N$ is an oriented submanifold of $N$ of dimension $k$. Moreover $\bar{Y} -Y$ is contained inside a submanifold in $N$ of dimension $k-2$ or less. Moreover $f :X \rightarrow Y$ is one to one and orientation preserving (but once extended to the closure, it may not be one to one). Does it imply that on the level of homology $$ f_*[\bar{X}] = [\bar{Y}] \in H_k(N, \mathbb{Z}) ?$$

Moreover generally, if this map was $r$ to one and restricted to a neighborhood of each point, the map is orientation preserving (only restricted to $X$), does it imply that

$$ f_*[\bar{X}] = r[\bar{Y}] \in H_k(N, \mathbb{Z}) ?$$

Note that, since the singular points of $\bar{X}$ and $\bar{Y}$ are of real codimension two or more, they define a homology class.

The particular example I have in mind is, when $M$ and $N$ are smooth compact algebraic varieties and $X$ and $Y$ are smooth subvarieteies and $f$ is a homolorphic map. Then the boundary of their closures will have real co-dimension two and hence, they automatically define homology classes. Everything is over the complex numbers.