Euler Classes and Numbers| 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) ∈ Hn(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 pm(P),
the corresponding Euler class is determined, non-uniquely, by e(P) ∧ e(P)
= pn/2(P).
> Online resources:
see Encyclopedia of Mathematics page;
Wikipedia page.
Euler Numbers or Characteristics > s.a. gauss-bonnet theorem.
$ Def: The Euler characteristic of a
d-complex C is χ(C):= ∑i
= 0d
(−1)i Ni(C),
where Ni(C) is the number
of i-faces of C.
$ Def: The Euler number
of an n-dimensional manifold M is defined as
χ(M):= ∫ e(F) .
* Relationships: It turns out that, in terms of Betti numbers,
χ(M) = ∑i = 0n (−1)i dim Hi(M; \(\mathbb R\)) = ∑i = 0n (−1)i bi .
* 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 indn(v)
= ind(v), where v is any vector field
which points outward on the boundary ∂M
(if any), with a finite number of zeroes xn,
and indn(v) the index of v at x.
* Odd-dimensional
manifolds: If M is a closed, odd-dimensional manifold, χ(M) = 0.
* Spheres: χ(Sn)
= 2 for n even, = 0 for n odd; χ(\(\mathbb C\)P2)
= χ(\(\mathbb C\)P3)
= 3; χ(K3) = 24.
* 1D manifolds: For an interval, χ(I) = 1; For a circle,
χ(S1) = 0.
* 2D manifolds:
For the tangent bundle of a manifold M, e(TM)
= −(1/2π) F12 ;
The Euler characteristic for a 2-sphere with g handles,
χ(M2g)
= 2 − 2g (2 for a 2-sphere, 0 for a torus, –2 for a double torus, etc);
For a disk, χ(D2) = 1;
χ(\(\mathbb R\)P2) = 1;
If M is a closed 2D manifold with metric,
χ(M) = \(1\over4\pi\)∫ R d2v .
* 4D manifolds: The Euler class of the tangent bundle of a manifold M is e(TM) = (1/32π2) εijkl R ik ∧ R jl ; The Euler characteristic for an S2-bundle over S2, χ = 4; For the product of two 2-manifolds, χ(M1 × M2) = 4 (1−g1) (1−g2); If M is a 4D manifold with metric,
χ(M) = \(1\over128\pi^2\)∫ Rabcd Refgh εabef εcdgh d4v .
@ 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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