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M is an n-dimensional closed orientable manifold. I find in a book "Intuitively,the fundamental class can be thought of as the sum of the (top-dimension) simplices of a suitable triangulation of the manifold". My explanation is we can use the Mayer-vietoris sequence to "glue" the local orientations inductively over a finite oriented atlas together to construct the fundamental class. One open subset of the atlas correspond to one simplex. Am I right?

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Yes, that's exactly how it works. The triangulation need not be "suitable", any triangulation works. You have to use signs if the triangulation is not compatible with the manifold's orientation but that's the only issue. –  Ryan Budney May 21 '13 at 19:54
    
Not exactly. In general the number of top dimensional simplices has to be higher than the number of open maps of an atlas. First of all you need to have any triangulation. If you do then any triangulation will be fine, and Mayer-Vietoris would not bother with this issue. Fix an orientation in one simplex, and it will spread over the entire connectivity component (over the whole manifold if it is connected; otherwise choose orientation for one simplex per component). –  Wlodzimierz Holsztynski May 22 '13 at 8:27
    
@Wlodzimierz: It's unclear what you are referring to, both with your "not exactly" comment and the "number of open maps of an atlas" comment. –  Ryan Budney May 22 '13 at 13:41
    
@Ryan: In the Question @jiangsaiyin seems to claim that there always can be a 1-1 correspondence between the open maps of an atlas and the simplices. That's not quite true. Also the mentioning of Mayer-Vietoris sounds not clear, more like unnecessary. On the other hand IF a triangulation of a compact manifold is given then one may hope, somewhat naively but why not, that each top simplex can be expanded to a map (just make them small enough to get different maps for different simplices), and indeed we would get a 1-1 correspondence between maps and the top simplices. –  Wlodzimierz Holsztynski May 22 '13 at 20:25
    
@jiangsaiyin: sphere $S^n$ can be covered by an atlas consisting of 2 maps, but for every positive $n$ the minimal number of $n$-simplices of a triangulation of $S^n$ is higher, is $n+2$. –  Wlodzimierz Holsztynski May 22 '13 at 20:29

2 Answers 2

Here is how I would prove it. You need to know two things. 1. If $M$ is connected, then $H_n (M) \to H_n (M,M-x)$ is an isomorphism (proven in any appropriate textbook). 2. If $\sigma:\Delta^n \to R^n$ is an embedding that has $o$ as an interior point of its image, then it represents a generator of $H_n (R^n,R^n - o)$. This is easily reduced to showing that the identity map on $\Delta^n$ represents a generator of $H_n (\Delta^n, \Delta^n - o)=H_n (\Delta^n, \partial \Delta^n)$, where $o$ is an interior point. By long exact sequence and excision, $H_n (\Delta^n, \partial \Delta^n) \to H_{n-1} (\partial \Delta^n) = H_{n-1}(\partial \Delta^n, \Lambda^n) \cong H_{n-1} (\Delta^{n-1}, \partial \Delta^{n-1})$, where $\Lambda^n$ is the horn, i.e. the union of all but one boundary simplices. By induction, the claim follows.

To get back to the manifold: pick a point $x$ in the interior of $\sigma$, one of the top-dimensional simplices in your triangulation. All that remains to be proven is that the sum of all top-dimensional simplices maps to $\sigma$ under $H_n (M) \to H_n (M,M-x)$, but that follows directly from what I wrote.

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Assuming you mean a smooth manifold, taking the top exterior power of the tangent bundle at every point gives you the determinant bundle, which is trivial when the manifold is orientable. Therefore you get a global nowhere vanishing section, which is an orientation. Along the same lines, one can produce a nowhere vanishing top-dimensional "volume" form which of course defines a fundamental class in de Rham cohomology. These are all correct "intuitions" but in your context you don't seem to be working with de Rham cohomology but rather with some version of singular (or simplicial) homology. It seems to me that your intuitions are all correct though of course without exact definitions it is hard to be sure. Certainly keeping de Rham's theorem in mind clarifies the "global" picture.

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