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(EF) = 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).

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(AB) = χ(A) + χ(B) – χ(AB) ,   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.

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 ikR 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].