# Euler characteristics and characteristic classes for real manifolds?

Consider an oriented manifold $X$. To calculate its Euler characteristic, one might integrate the Euler class. Now if $X$ were a complex manifold, and given as a section of some complex bundle $E$ over $Y$, we would have

$\chi(X) = \int\limits_X \mbox{ } c_* (TY) / c_* (E)$

where $c_*(\cdot)$ is the total Chern class.

Can anything of the sort be said if $X$ is a real manifold?

Presumably, if one wants only the Euler characteristic modulo 2, one can use the Stiefel-Whitney classes instead of the Chern classes. On the other hand, it seems to me that the topology of $TY$ and $E$ as bundles over $Y$ cannot suffice to carry the information of the Euler characteristic of the zero locus of a section of $E$. So I guess what I'm really asking is:

What should I know about a section $\sigma:Y\to E$ in order to know the Euler number of its intersection with the zero section, assuming this is transverse?

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Perhaps you meant "... a section $\sigma:Y\to E$ ..."? – Somnath Basu May 21 '11 at 2:42
Your question isn't clear to me. What does "real manifold" mean? And "Can anything of the sort be said..." of course, but what sort of thing are you interested in? – Ryan Budney May 21 '11 at 4:04
It's noteworthy that for a rank $r$ oriented bundle $E$ on an $n$-dimensional compact oriented manifold $Y$ the Euler characteristic of the zero locus $X$ of a transverse section is given by integrating the Euler class of $TY$ over $X=Y$ in the extreme case $r=0$, and by integrating the Euler class of $E$ over the zero-dimensional oriented manifold $X$ in the other extreme case $r=n$. When $0<r<n$ I can't imagine what an answer could look like. If $n=4$ and $r=2$ then the Euler characteristic of $X$ can be any even integer, for any choice of $Y$ and $E$. – Tom Goodwillie May 21 '11 at 4:05
@Sommath, yes, thanks. @Ryan, I just meant to emphasize it wasnt a complex manifold. – Vivek Shende May 21 '11 at 6:17
Vivek -- a small remark: the Euler characteristic of the mod 2 cohomology coincides with the Euler characteristic of the cohomology with any field coefficients and with the Euler characteristic of the integral cohomology (defined as the alternating sum of the ranks). – algori May 21 '11 at 13:39

As far as I understand the question, it is asking: given a(n oriented) vector bundle $E$ on a(n oriented) manifold, what information on the Euler characteristic of the zero locus of a transversal section of $E$ can we deduce from the characteristic classes of $E$?
I'm afraid the answer to that is "in general, not much, apart from the parity". For instance, every orientable surface $S$ is the zero locus of a function (i.e., a section of the trivial 1-bundle) on the 3-sphere: embed the surface in the sphere and take the "oriented distance" function. In a similar way one can realize $S$ as the zero locus of two functions on $S^4$: realize $S^3$ as the equator in $S^4$, take one of the functions to be the oriented distance function extended to $S^4$ and the other a height function whose zero locus is $S^3$.