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.