[ 3 / biz / cgl / ck / diy / fa / ic / jp / lit / sci / vr / vt ] [ index / top / reports ] [ become a patron ] [ status ]
2023-11: Warosu is now out of extended maintenance.

/sci/ - Science & Math

View post   
File: 111 KB, 680x918, ec.png [View same] [iqdb] [saucenao] [google]
8844465 No.8844465 [Reply] [Original]

is there an intuitive descritption of Euler characteristic? alternating sum of Betti numbers doesn't have any sense to me

>> No.8844531

I always think about this scenario when I hear about Euler:

Euler rubbing me with oil and after he finishes I say to him "You're a good oiler, Euler"

>> No.8844543
File: 12 KB, 243x243, 1492513170302.png [View same] [iqdb] [saucenao] [google]
8844543

>>8844531

jesus christ what a (shit)post

>> No.8844545

>>8844543
I just wanted to bump this thread in a different way than to just write "bump".

>> No.8844549

>>8844465
In function theory we defined it like this.
Define the pth standard simplex as the convex hull of
(0,0) p=0
(0,0) , (1,0) p=1
(0,0) , (1,0) , (0,1) p=2
then we can triangulate that meaning we divide a topolical space by homöomorphisms of this standard simplex. There we get the the image of the 0th standard simplex as the corners. The image of the 1th as the interior of the connection of two corners meaning an edge and the image of the 2th as the interior of a side (a surface).
Now the Euler characteristics is defined as the number of areas - number of edges + number of corners. Therefore it's also a topological invariant between to conform equivalent spaces.

>> No.8844550
File: 4 KB, 210x230, 1492780214782.jpg [View same] [iqdb] [saucenao] [google]
8844550

>>8844545

thanks, you really DID make me think

>> No.8844573

>>8844549
>number of areas - number of edges + number of corners

ok, it's a topological inveriant, but does that number mean something else?

>> No.8844588

>>8844573
Look up its relation to vector fields on manifolds.

>> No.8844592

>>8844549
I forgot to say that this is only for compact orientable spaces and independent of the triangulation
>>8844573
I don't think so. One can define the genus with it which is important for the Riemann mapping theorem on compact Riemann surfaces.

>> No.8844594

>>8844588 may probably know more than me

>> No.8844634

>>8844465
This is a neat chart. Where can I find more charts like this?

>> No.8844641
File: 80 KB, 766x960, 1453317220469.jpg [View same] [iqdb] [saucenao] [google]
8844641

>>8844634
it's literally from wikipedia

>> No.8844673

>>8844465
Nice rigorous proof for the sphere is: take a tetrahedron and blow it up. Now you have a sphere. Now you can calculate it as 4 areas - 6 edges + 4 corners = 2. For the torus you can do it similar to get 0, but more difficult. Then you can get a body with every negative even Euler characterstic by adding a torus to a sphere.

>> No.8844676

>>8844588

too much of a brainlet to figure it out quickly, I definitively look into it.

reading this paper now http://www.crm.cat/en/Publications/Publications/2008/Preprints/Pr793.pdf

I feel like some deep combinatiorial meaning is possible. In the paper above they call it a generalized cardinality