Euler class

In mathematics , more precisely in algebraic topology and in differential geometry and topology , the Euler class is a special type of characteristic class that is assigned to orientable real vector bundles . It is named after Leonhard Euler because in the case of the tangential bundle of a manifold it determines its Euler characteristic .

It can be defined in different (equivalent) ways: as an obstacle to the existence of a cut without zeros, as a pull-back of the orientation class under a cut or as an image of the Pfaff determinant under the Chern-Weil isomorphism. In the case of shallow bundles, there are other equivalent definitions.

Basic idea and motivation

The Euler class is a characteristic class , i.e. a topological invariant of oriented vector bundles: two isomorphic oriented vector bundles have the same Euler classes. In the case of differentiable manifolds , the Euler class of the tangent bundle determines the Euler characteristic of the manifold.

The Euler class provides an obstacle to the existence of a cut without zeros. In particular, the Euler characteristic of a closed, orientable, differentiable manifold provides an obstacle to the existence of a vector field without singularities.

For a zero-point cut defined on a subset of the base space, one can define a relative Euler class; this provides an obstacle to the continuation of the cut without zero points on the entire base.

Axioms

The (relative) Euler class is determined by the following axioms.

Every oriented, -dimensional real vector bundle with a nowhere vanishing intersection on a (possibly empty) subset becomes an element ${\displaystyle n}$ ${\displaystyle E\to X}$ ${\displaystyle s\colon Y\to E}$ ${\displaystyle Y\subset X}$

${\displaystyle e(E,s)\in H^{n}(X,Y;\mathbb {Z} )}$

(or if ) assigned so that ${\displaystyle e(E)\in H^{n}(X;\mathbb {Z} )}$ ${\displaystyle Y=\emptyset }$

• holds for every continuous mapping ${\displaystyle f\colon (X^{\prime },Y^{\prime })\to (X,Y)}$ ${\displaystyle e(f^{*}E,f^{*}s)=f^{*}(e(E,s))}$
• ${\displaystyle e(E_{1}\oplus E_{2},s\oplus 0)=e(E_{1},s)\cup e(E_{2})}$
• for the tautological complex line bundle , understood as a 2-dimensional real vector bundle, is a generator of . ${\displaystyle \gamma ^{1}\to \mathbb {C} P^{1}}$ ${\displaystyle e(\gamma ^{1})\in H^{2}(\mathbb {C} P^{1};\mathbb {Z} )}$ ${\displaystyle H^{2}(\mathbb {C} P^{1};\mathbb {Z} )\simeq \mathbb {Z} }$

${\displaystyle e(E)\in H^{n}(X;\mathbb {Z} )}$is called the Euler class of the bundle , is called the relative Euler class relative to the cut . ${\displaystyle E}$ ${\displaystyle e(E,s)\in H^{n}(X,Y;\mathbb {Z} )}$ ${\displaystyle s}$

Definition as an obstruction class

For a -dimensional vector bundle oriented to the geometric realization of a simplicial complex is obtained by means of obstruction theory the obstruction class ${\displaystyle n}$ ${\displaystyle E\to \vert K\vert }$ ${\displaystyle \vert K\vert }$ ${\displaystyle K}$

${\displaystyle o_{n}(E)\in H^{n}(K;\pi _{n-1}(V_{1}(\mathbb {R} ^{n}))}$

for the continuation of a cut in the associated vector bundle onto the skeleton of . ${\displaystyle n}$ ${\displaystyle K}$

The coefficient group

${\displaystyle \pi _{n-1}(V_{1}(\mathbb {R} ^{n}))\simeq \pi _{n-1}(\mathbb {R} ^{n}-0)\simeq H_{n-1}(\mathbb {R} ^{n}-0;\mathbb {Z} )\simeq H_{n}(\mathbb {R} ^{n},\mathbb {R} ^{n}-0;\mathbb {Z} )}$

is canonically isomorphic to (due to the orientation) and this isomorphism maps to the Euler class . ${\displaystyle \mathbb {Z} }$ ${\displaystyle o_{n}(E)}$ ${\displaystyle e(E)\in H^{n}(K;\mathbb {Z} )}$

Definition by means of an orientation class

For an oriented -dimensional vector bundle and the complement of the zero cut we consider the image of the orientation class (Thom class) ${\displaystyle n}$ ${\displaystyle p\colon E\to M}$ ${\displaystyle E_{0}\subset E}$ ${\displaystyle u\mid _{E}}$

${\displaystyle u\in H^{n}(E,E_{0};\mathbb {Z} )}$

in . Because is contractible , is a homotopy equivalence and ${\displaystyle H^{n}(E;\mathbb {Z} )}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle p\colon E\to M}$

${\displaystyle p^{*}\colon H^{*}(M;\mathbb {Z} )\to H^{*}(E;\mathbb {Z} )}$

an isomorphism . The Euler class is defined by

${\displaystyle e(E):=(p^{*})^{-1}u\mid _{E}\in H^{n}(M;\mathbb {Z} )}$.

You can get equivalent through ${\displaystyle e(E)}$

${\displaystyle e(E):=s^{*}u\mid _{E}}$

Define for any section (e.g. the zero section) . ${\displaystyle s\colon M\to E}$

If has an intersection without zeros, that is, it follows . ${\displaystyle E\to M}$ ${\displaystyle s(M)\subset E_{0}}$ ${\displaystyle e(E)=0}$

Relative Euler class : If there is an intersection without zeros on a subset , then it can be continued to an intersection (possibly with zeros) and then defined ${\displaystyle s_{0}\colon Y\to E}$ ${\displaystyle Y\subset M}$ ${\displaystyle s\colon (M,Y)\to (E,E_{0})}$

${\displaystyle e(E,s_{0}):=s^{*}u\in H^{n}(M,Y;\mathbb {Z} )}$.

Definition via Chern-Weil theory

We consider vector bundles over a differentiable manifold . The construction using the Chern-Weil theory (only) provides the image of the Euler class in or the relative Euler class in , in particular it provides the zero class for vector bundles of odd dimensions. ${\displaystyle M}$ ${\displaystyle H^{*}(M;\mathbb {R} )}$ ${\displaystyle H^{*}(M,Y;\mathbb {R} )}$

For an oriented vector bundle of dimension , consider the associated principal bundle (the frame bundle ) . ${\displaystyle n=2k}$ ${\displaystyle SO(2k)}$ ${\displaystyle P\to M}$

For a principle bundle with a connected form , the Euler class is the image of through ${\displaystyle SO(2k)}$ ${\displaystyle P\to M}$ ${\displaystyle \omega \in \Omega ^{1}(P,so(2k))}$ ${\displaystyle e(P)\in H_{dR}^{2k}(M)\simeq H^{2k}(M;\mathbb {R} )}$

${\displaystyle Pf(A,\ldots ,A)={\frac {1}{2^{k}k!}}\sum _{\sigma \in S_{2k}}sign(\sigma )a_{\sigma (1)\sigma (2)}\ldots a_{\sigma (2k-1)\sigma (2k)}}$

defined Pfaff's determinant under the Chern-Weil homomorphism ${\displaystyle Pf\in I^{n}(so(2k))}$

${\displaystyle I^{k}(so(2k))\to H_{dR}^{2k}(M)}$,

that is, the differential shape defined with the help of the curvature shape of the principal bundle ${\displaystyle \Omega \in \Omega ^{2}(M)}$

${\displaystyle {\frac {1}{2^{k}\pi ^{2k}}}Pf(\Omega )(X_{1},\dots ,X_{2k}):={\frac {1}{(2\pi )^{2k}}}{\frac {1}{(k)!}}\sum _{\sigma \in {\mathfrak {S}}_{2k}}\operatorname {sign} (\sigma )Pf(\Omega (X_{\sigma (1)},X_{\sigma (2)}),\dots ,\Omega (X_{\sigma (2k-1)},X_{\sigma (2k)}))}$

represented De Rham cohomology class. One can show that the Euler class does not depend on the choice of the form of connection and that it lies in the image of . ${\displaystyle \Omega }$ ${\displaystyle H^{2k}(M;\mathbb {Z} )}$

The correspondence of the Euler class defined in this way with the topologically defined above is the content of the generalized Gauss-Bonnet theorem proved by Allendoerfer and Weil in 1943 (and with an intrinsic proof in 1944 by Chern) .

Relative Euler class : Let it be an intersection without zeros over a submanifold . (We assume that the cut can be continued to an open neighborhood of .) Then there is a connected form whose curvature form satisfies. In particular, defines a relative cohomology class . ${\displaystyle s\colon Y\to E}$ ${\displaystyle Y\subset M}$ ${\displaystyle Y}$ ${\displaystyle \omega }$ ${\displaystyle Pf(\Omega )\mid _{Y}\equiv 0}$ ${\displaystyle Pf(\Omega )}$ ${\displaystyle e(E,s)\in H^{2k}(M,Y;\mathbb {Z} )}$

Euler class of SL (n, R) principal bundles

Among the isomorphisms

${\displaystyle I^{k}(so(2k))\simeq H^{2k}(BSO(2k))\simeq H^{2k}(BSL(2k,\mathbb {R} ))}$

the Pfaff determinant corresponds to a cohomology class in the cohomology of the classifying space , the Euler class of the universal bundle . The Euler class can be assigned to each bundle using the classifying mapping ${\displaystyle e(\gamma ^{2k})}$ ${\displaystyle BSL(2k,\mathbb {R} )}$ ${\displaystyle \gamma ^{2k}\to BSL(2k,\mathbb {R} )}$ ${\displaystyle SL(2k,\mathbb {R} )}$ ${\displaystyle P\to M}$ ${\displaystyle f\colon M\to BSL(2k,\mathbb {R} )}$

${\displaystyle e(P):=f^{*}(e(\gamma ^{2k}))\in H^{2k}(M)}$

define. This corresponds to the Euler class of the associated vector bundle.

Euler class of bundles of spheres

The Euler class can be defined for any bundle of spheres .

In the case of the unit sphere bundle of a Riemann vector bundle , the Euler class of the vector bundle defined above is obtained.

properties

• The canonical homomorphism maps the Euler class to the nth Stiefel-Whitney class . ${\displaystyle H^{n}(X;\mathbb {Z} )\to H^{n}(X;\mathbb {Z} /2\mathbb {Z} )}$ ${\displaystyle w_{n}\in H^{n}(X;\mathbb {Z} /2\mathbb {Z} )}$
• The cup product is the highest Pontryagin class . ${\displaystyle e(E)\cup e(E)}$ ${\displaystyle p_{n}\in H^{2n}(X;\mathbb {Z} )}$
• For closed , orientable , differentiable manifolds with tangent and fundamental class is the Euler characteristic of . ${\displaystyle M}$ ${\displaystyle TM}$ ${\displaystyle \left[M\right]}$ ${\displaystyle \langle e(TM),\left[M\right]\rangle =\chi (M)}$ ${\displaystyle M}$
• Let it be the vector bundle with the opposite orientation, then is . ${\displaystyle {\overline {E}}}$ ${\displaystyle E}$ ${\displaystyle e({\overline {E}})=-e(E)}$
• In particular, we have for vector bundles of odd dimension . The Euler characteristic vanishes for closed, orientable, differentiable manifolds of odd dimensions. ${\displaystyle 2e(E)=0}$
• For the Whitney sum and the Cartesian product of vector bundles we have

  ${\displaystyle e(E_{1}\oplus E_{2})=e(E_{1})\cup e(E_{2}),e(E_{1}\times E_{2})=e(E_{1})\times e(E_{2}),}$
  where denotes the cup product and the cross product . ${\displaystyle \cup }$ ${\displaystyle \times }$

• For a generic intersection of a -dimensional oriented vector bundle over a -dimensional closed orientable manifold , the image of the fundamental class is the set of zeros in the Poincaré dual of . In the case of the tangential bundle, this gives the Poincaré-Hopf theorem . ${\displaystyle s\colon M\to E}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle \left[Z\right]}$ ${\displaystyle Z=\left\{x\in X:s(x)=0\right\}}$ ${\displaystyle H_{m-n}(M;\mathbb {Z} )}$ ${\displaystyle e(E)\in H^{n}(M;\mathbb {Z} )}$ ${\displaystyle E=TM}$
• If the normal bundle is a closed orientable submanifold , then the self intersection number is from . ${\displaystyle N_{Y}}$ ${\displaystyle Y\subset M}$ ${\displaystyle <e(N_{Y}),\left[Y\right]>}$ ${\displaystyle Y}$
• If an intersection is without zeros, then it is for everyone . ${\displaystyle s\colon X\to E}$ ${\displaystyle e(E,s\vert _{Y})=0\in H^{n}(X,Y;\mathbb {Z} )}$ ${\displaystyle Y\subset X}$
• Gysin sequence : For a-dimensional oriented vector bundle(withthe set of non-zero vectors) the cup product with the Euler class conveys an exact sequence ,the other two imagesand the integration being along the fiber . ${\displaystyle n}$ ${\displaystyle E\to B}$ ${\displaystyle E_{0}\subset E}$
  ${\displaystyle \ldots \to H^{i}(B;\mathbb {Z} )\to H^{i+n}(B)\to H^{i+n}(E_{0})\to H^{i+1}(B)\to \ldots }$
  ${\displaystyle \pi \vert _{E_{0}}^{*}}$

Euler class flat bundle

Simplicial definition

Let it be a flat vector bundle over the geometric realization of a simplicial complex with -Simplices. Because simplices are contractible, the bundle is trivial over every simplex. For arbitrarily chosen one can thus construct a section by affine continuation . For generics , this intersection has no zeros on the skeleton, at most one zero per simplex and is transverse to the zero intersection. Then we define a simplicial -Cocycle through ${\displaystyle p\colon E\to \vert K\vert }$ ${\displaystyle \vert K\vert }$ ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle v_{1},\ldots ,v_{n}}$ ${\displaystyle s(v_{i})\in p^{-1}(v_{i})}$ ${\displaystyle s\colon \vert K\vert \to E}$ ${\displaystyle s(v_{i})}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle {\mathcal {E}}(\sigma )=0}$if has no zero ${\displaystyle s\vert _{\sigma }}$

${\displaystyle {\mathcal {E}}(\sigma )=1}$if a and if a positive basis of even a positive basis is ${\displaystyle s(p)=0}$ ${\displaystyle p\in \sigma }$ ${\displaystyle t_{1},\ldots ,t_{n}}$ ${\displaystyle T_{p}K}$ ${\displaystyle t_{1},\ldots ,t_{n},d_{p}s(t_{1}),\ldots ,d_{p}s(t_{n})}$ ${\displaystyle T_{s(p)}E}$

${\displaystyle {\mathcal {E}}(\sigma )=-1}$ otherwise.

One can show that there is a cocycle and that its value on cycles does not depend on the chosen section. The cohomology class represented by is the Euler class of the flat bundle. ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle e\in H^{n}(K;\mathbb {Z} )}$

Flat SL (2, R) bundles

Because of the universal superimposition ${\displaystyle \pi _{1}SL(2,\mathbb {R} )\simeq \mathbb {Z} }$

${\displaystyle \mathbb {Z} \to {\widetilde {SL(2,\mathbb {R} )}}\to SL(2,\mathbb {R} )}$,

this is a central extension and is therefore represented by a cohomology class. This is the universal Euler class for flat bundles, i.e. H. for a flat bundle with holonomy representation one obtains ${\displaystyle E\in H^{2}(BSL(2,\mathbb {R} )^{\delta };\mathbb {Z} )}$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle P\to M}$ ${\displaystyle \rho \colon \pi _{1}M\to SL(2,\mathbb {R} )}$

${\displaystyle e(P)=f^{*}(B\rho )^{*}(E)\in H^{2}(M;\mathbb {Z} )}$,

where is the classifying map of the universal overlay. ${\displaystyle f\colon M\to B\pi _{1}M}$

Flat circle bundles

It denotes the group of orientation-preserving homeomorphisms of the circle. Your universal overlay is . The integers work through translations to and obtained an exact sequence ${\displaystyle Homeo^{+}(S^{1})}$ ${\displaystyle {\widetilde {Homeo^{+}(S^{1})}}=\left\{f\colon \mathbb {R} \to \mathbb {R} :f(x+1)=f(x)+1\ \forall x\in \mathbb {R} \right\}}$ ${\displaystyle \mathbb {R} }$

${\displaystyle \mathbb {Z} \to {\widetilde {Homeo^{+}(S^{1})}}\to Homeo^{+}(S^{1})}$.

The associated group cohomology class is the universal Euler class for flat bundles. ${\displaystyle E\in H^{2}(Homeo^{+}(S^{1});\mathbb {Z} )}$ ${\displaystyle Homeo^{+}(S^{1})}$

An explicit formula was given by Jekel: the universal Euler class is represented by the so-called orientation cocycle : ${\displaystyle E_{\mathbb {R} }\in H^{2}(Homeo^{+}(S^{1});\mathbb {R} )}$ ${\displaystyle o\in C^{2}(Homeo^{+}(S^{1};\mathbb {R} )}$

${\displaystyle o(g_{0},g_{1},g_{2})={\frac {1}{2}}}$if arranged clockwise on the circle ${\displaystyle g_{0}(1),g_{1}(1),g_{2}(1)}$

${\displaystyle o(g_{0},g_{1},g_{2})=0}$if at least two of the values match ${\displaystyle g_{0}(1),g_{1}(1),g_{2}(1)}$

${\displaystyle o(g_{0},g_{1},g_{2})=-{\frac {1}{2}}}$if they are arranged counterclockwise on the circle. ${\displaystyle g_{0}(1),g_{1}(1),g_{2}(1)}$

The orientation cocycle then also represents the universal Euler class for flat bundles for all subgroups . This is especially true for flat bundles: use the effect of on through fractional-linear transformations . ${\displaystyle G\subset Homeo^{+}(S^{1})}$ ${\displaystyle G}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle S^{1}=P^{1}\mathbb {R} }$

literature

• John W. Milnor, James D. Stasheff: Characteristic classes . In: Annals of Mathematics Studies , No. 76. Princeton University Press, Princeton NJ; University of Tokyo Press, Tokyo 1974. (Chapter 9)
• Johan L. Dupont: Curvature and characteristic classes . In: Lecture Notes in Mathematics , Vol. 640. Springer-Verlag, Berlin / New York 1978, ISBN 3-540-08663-3
• Raoul Bott, Loring W. Tu: Differential forms in algebraic topology . In: Graduate Texts in Mathematics , 82nd Springer-Verlag, New York / Berlin 1982, ISBN 0-387-90613-4 (Chapter 11)
• Riccardo Benedetti, Carlo Petronio: Lectures on hyperbolic geometry. University text. Springer-Verlag, Berlin 1992, ISBN 3-540-55534-X (Chapter F.4)
• Tammo tom Dieck: Algebraic topology. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zurich 2008, ISBN 978-3-03719-048-7 (Chapter XI)
• Alberto Candel, Lawrence Conlon: Foliations. II . In: Graduate Studies in Mathematics , 60th American Mathematical Society, Providence RI 2003, ISBN 0-8218-0881-8 (Chapter 4)

Web links

• Vladimir Sharafutdinov: Relative Euler class and the Gauß-Bonnet theorem (PDF)
• Michelle Bucher-Karlsson: Characteristic classes and bounded cohomology . (PDF; Chapter 3.1)
• Yin Li: The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds . arxiv : 1111.4972

Individual evidence

1. ↑ Milnor-Stasheff (op.cit.), Theorem 12.5
2. ↑ Shiing-Shen Chern: On the curvatura integra in a Riemannian manifold . In: Annals of Mathematics , 46 (4), 1945, pp. 674-684.
3. ↑ Sharafutdinov (op.cit.), Chapter 2
4. ↑ Bott-Tu (op.cit.), Chapter 11
5. ↑ Benedetti-Petronio (op.cit.), Lemma F.4.1
6. ↑ Benedetti-Petronio (op.cit.), Lemma F.4.2
7. ↑ Benedetti-Petronio (op.cit.), Proposition F.4.4 and F.4.3
8. ↑ Bucher-Karlsson (op.cit.), Section 3.1.4
9. ↑ Solomon M. Jekel: A simplicial formula and bound for the Euler class . In: Israel J. Math. , 66, 1989, no. 1-3, pp. 247-259.