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

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.

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It's not relevant to your example (where you left out "over $\mathbb C$"), but in general that $r$ has to be a sum of $\pm 1$, depending on whether the map is locally orientation-preserving or -reversing. – Allen Knutson Jun 8 '13 at 5:15
Thank you for pointing out the mistake. I have edited the question. And regarding the special case I have in mind, I am indeed considering varieties over $\mathbb{C}$. – Ritwik Jun 8 '13 at 6:04
i am confused by title vs text, i.e. defn vs properties. are the "submanifolds" cx. analytic? then the closures of X and Y are *-analytic, so analytic. if M,N are projective, X,Y are algebraic by Chow. So homology is not needed for degree. Mumford, Alg. Geom I, pp.60,46. and M,N are always birational to projective vars. Sard's theorem lets one define degree in your setting, pointwise, as r. what do you want about "degree f"? behavior on homology? or # of preimages of a general point? i.e. both defns are ok - so to title, "yes". do you need them also to be equal, as in your text? – roy smith Jun 11 '13 at 16:28
If so, then Hironaka's the. on resolution of varieties and morphisms reduces the question to the cases of blowups and of regular maps of smooth varieties, where presumably it is known. cf Dold, lectures on alg top, p.268 for the smooth case. – roy smith Jun 11 '13 at 18:14
Regarding Roy Smith's comment: If $M$ and $N$ are projective, then $X$ and $Y$ are algebraic, but is the closure necessarily algebraic? This is the content of this discussion: mathoverflow.net/questions/119848/… What I want about degree is the behavior on Homology. But the hypothesis/settings is that I have a map that is $k$ to one and orientation preserving restricted to $X$. So it would probably be helpful to think of degree as number of preimages of a regular value, counted with a sign. – Ritwik Jun 12 '13 at 3:32

## 2 Answers

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.

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That is correct, the spaces $\bar{X}$ and $\bar{Y}$ are indeed pseudomanifolds. Is it a known fact about pseudomanifolds, the degree'' of a map? Do you know of some reference? – Ritwik Jun 8 '13 at 15:09
references added. – BS. Jun 10 '13 at 9:26

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.

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