Euler Classes and Numbers |

**Euler Classes**

* __Idea__: Characteristic classes, used for oriented,
real vector bundles with structure group *G* = SO(*k*), with *k* even, which, like
other characteristic classes, measures how "twisted" the vector bundle is.

* __Notation__: They are denoted by
*e*(*P*) ∈ H^{n}(*B*; \(\mathbb R\)),
for even *n*.

* __Whitney sum__: *e*(*E* ⊕ *F*)
= *e*(*E*) *e*(*F*) (in terms of forms, this means exterior product).

* __Relationships__: Given
a Pontrjagin class *p _{m}*(

>

**Euler Numbers or Characteristics** > s.a. gauss-bonnet theorem.

$ __Def__: The Euler characteristic of a
*d*-complex *C* is *χ*(*C*):= ∑_{i
= 0}^{d}
(−1)^{i} *N _{i}*(

$

*χ*(*M*):= ∫ *e*(*F*) .

* __Relationships__: It turns
out that, in terms of Betti numbers,

*χ*(*M*) = ∑_{i =
0}^{n} (−1)^{i}
dim H^{i}(*M*; \(\mathbb R\))
= ∑_{i =
0}^{n}
(−1)^{i} *b _{i}* .

* __Relationship with operations on manifolds__:
For the union, Cartesian product and connected sum of manifolds, respectively,

*χ*(*A* ∪ *B*)
= *χ*(*A*) + *χ*(*B*)
− *χ*(*A* ∩ *B*) , *χ*(*A*
× *B*) = *χ*(*A*) *χ*(*B*) ,
*χ*(*A* # *B*)
= *χ*(*A*) + *χ*(*B*)
− [1+(−1)^{n}] .

* __Applications__: It classifies
2D spaces, but is very weak in higher dimensions; It is used to give a condition
for the existence of a Lorentzian metric on a compact manifold.

@ __References__: Borsten et al a2105 [odd-dimensional analog].

> __Online resources__: see
Wikipedia page.

**Examples**

* __Compact manifold__: *χ*(*M*)
= ∑* _{n}* ind

*

*

*

*

*χ*(*M*) = \(1\over4\pi\)∫
*R* d^{2}*v* .

* __4D manifolds__:
The Euler class of the tangent bundle of a manifold *M* is *e*(T*M*)
= (1/32π^{2})
*ε _{ij}*

*χ*(*M*) = \(1\over128\pi^2\)∫ *R _{abcd}*

@ __Special cases__:
Roček & Williams PLB(91) [piecewise linear].

@ __Related topics__: Eastwood & Huggett EJC(07)
[family of manifolds whose Euler characteristics are related to the chromatic polynomial of a graph].

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