PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
Euler characteristic (Definition)

The term Euler characteristic is defined for several objects.

If $ K$ is a finite simplicial complex of dimension $ m$, let $ \alpha_i$ be the number of simplexes of dimension $ i$. The Euler characteristic of $ K$ is defined to be

$\displaystyle \chi(K) = \sum_{i=0}^m (-1)^i \alpha_i . $

Next, if $ K$ is a finite CW complex, let $ \alpha_i$ be the number of i-cells in $ K$. The Euler characteristic of $ K$ is defined to be

$\displaystyle \chi(K) = \sum_{i \ge 0}(-1)^i \alpha_i . $

If $ X$ is a finite polyhedron, with triangulation $ K$, a simplicial complex, then the Euler characteristic of $ X$ is $ \chi(K)$. It can be shown that all triangulations of $ X$ have the same value for $ \chi(K)$ so that this is well-defined.

Finally, if $ C=\{C_q\}$ is a finitely generated graded group, then the Euler characteristic of $ C$ is defined to be

$\displaystyle \chi(C) = \sum_{q \ge 0} (-1)^q rank(C_q) . $



"Euler characteristic" is owned by Mathprof.
(view preamble)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: group, finitely generated, well-defined, triangulation, finite polyhedron, CW complex, dimension, simplicial complex, finite, objects, term
There are 13 references to this entry.

This is version 10 of Euler characteristic, born on 2006-09-02, modified 2007-05-24.
Object id is 8311, canonical name is EulerrCharacteristic.
Accessed 1715 times total.

Classification:
AMS MSC: 55N99 (Algebraic topology :: Homology and cohomology theories :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)