Is there an analogous concept for the degree of a map, when the spaces are singular?

Question

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.

Answer 1

It seems that what you are after is the concept of pseudomanifold.

EDIT: A suitable reference might be Massey's "A basic course in algebraic topology" (Springer 1991), chapter IX §8, where it is shown that an orientable $n$-dimensional pseudomanifold has infinite cyclic $n$-th homology group. Beware that his definition of pseudomanifold (in terms of regular CW complexes) assumes "irreducibility", namely that the complement of the singular set is connected. There are also exercises in §43 of Munkres' "Elements of algebraic topology" to the same effect, although the definition is in terms of simplicial complexes.

Then any continuous map between oriented $n$-dimensional pseudomanifolds has a degree, given by its action on $n$-dimensional homology.

note that to apply this to (singular, compact, irreducible) complex varieties, one must resort to the highly non-trivial result (due to Whitney, 1938) that they are triangulable.

This has been vastly generalized by Goresky-MacPherson with their intersection homology theory, giving rise to a kind of Poincaré duality for singular spaces.

Answer 2

I should answer your question as it is titled. Here is an example where the degree of a map between a manifold and an orbifold is defined. Let f from X (a connected orientable manifold ) to X/G be the canonical projection. The group G is finite, it acts faithfully on X and preserves the orientation. Then f has a degree equal to the order of G. e.g. consider the well-known map f : P^n --> P^n(Q), between usual projective space and weighted projective space ( over C, dimension n ). Then f has a degree equal to the product of the weights in Q divided by their gcd.