Milnor-Wood inequality

In the mathematical field of differential geometry , the Milnor-Wood inequality gives an obstacle to the existence of a flat relationship on a fiber bundle .

Theorems of Milnor and Wood

The classic Milnor-Wood inequality concerns (oriented) flat bundles of circles over a surface , the monodromy of which is given by a homomorphism . A stronger inequality is obtained for linear flat bundles of circles, i.e. with monodromy ${\displaystyle \Sigma }$ ${\displaystyle \rho \colon \pi _{1}\Sigma \to Homeo^{+}(S^{1})}$ ${\displaystyle \rho \colon \pi _{1}\Sigma \to SL(2,\mathbb {R} )}$

Sentence (Milnor): Be a closed , orientable surface of gender . For the Euler class of a linear flat circle bundle over holds ${\displaystyle \Sigma }$ ${\displaystyle g}$ ${\displaystyle \pi \colon E\to \Sigma }$ ${\displaystyle \Sigma }$

${\displaystyle \vert eu(E)\vert \leq g-1}$.

In particular, the tangential bundle of a closed, orientable surface does not have a flat relationship with the sex . ${\displaystyle g\geq 2}$

Theorem (Wood): Be a closed, orientable surface of the gender . A flat circular bundle for the Euler class above applies ${\displaystyle \Sigma }$ ${\displaystyle g}$ ${\displaystyle \pi \colon E\to \Sigma }$ ${\displaystyle \Sigma }$

${\displaystyle \vert eu(E)\vert \leq 2g-2}$.

Milnor's theorem follows from Wood's theorem, because for a representation by means of the 2-fold superposition one gets a representation whose Euler number is exactly half the Euler number of the original representation. ${\displaystyle \pi _{1}\Sigma \to SL(2,\mathbb {R} )}$ ${\displaystyle SL(2,\mathbb {R} )\to PSL(2,\mathbb {R} )}$ ${\displaystyle \pi _{1}\Sigma \to PSL(2,\mathbb {R} )\subset Homeo^{+}(S^{1})}$

It follows from Goldman's Theorem that the representations with are exactly the most discrete and faithful representations . ${\displaystyle \rho \colon \pi _{1}\Sigma \to PSL(2,\mathbb {R} )}$ ${\displaystyle eu(E_{\rho })=\pm (2g-2)}$

Limited cohomology

The Milnor and Wood proofs used estimates for commutators in and . A simpler proof going back to Ghys and Jekel uses bounded cohomology : the universal Euler class in can be represented by a bounded cocycle of norm 1/2. ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle Homeo^{+}(S^{1})}$ ${\displaystyle H^{2}(Homeo^{+}(S^{1});\mathbb {R} )}$

Generalizations

Higher dimensional bundles

Theorem (Sullivan-Smillie): For the Euler class of a flat -bundle over a closed, orientable manifold the inequality holds ${\displaystyle GL^{+}(n,\mathbb {R} )}$ ${\displaystyle \pi \colon E\to M}$ ${\displaystyle M}$

${\displaystyle \vert eu(E)\vert \leq {\frac {1}{2^{n}}}\parallel M\parallel }$

for the Euler class and the simplicial volume . ${\displaystyle eu(E)}$ ${\displaystyle \parallel M\parallel }$

One says that a manifold of the Milnor-Wood inequality with constant is sufficient if for every flat bundle the inequality ${\displaystyle M}$ ${\displaystyle MW(M)\in \mathbb {R} }$ ${\displaystyle GL^{+}(n,\mathbb {R} )}$

${\displaystyle \vert eu(E)\vert \leq MW(M)\vert \xi (M)\vert }$

holds for the Euler class and the Euler characteristic . It follows from Milnor's theorem and from a Bucher-Gelander theorem . ${\displaystyle eu(E)}$ ${\displaystyle \chi (M)}$ ${\displaystyle MW(\Sigma _{g})={\frac {1}{2}}}$ ${\displaystyle MW(M_{1}\times M_{2})=MW(M_{1})MW(M_{2})}$

Toledo invariant

Let be a Hermitian symmetric space of non-compact type of rank and the Kähler form of the Bergman metric . Then the inequality holds for a continuous mapping of a surface from gender ${\displaystyle X=G/K}$ ${\displaystyle p}$ ${\displaystyle \omega }$ ${\displaystyle f\colon \Sigma \to X}$ ${\displaystyle \Sigma }$ ${\displaystyle g}$

${\displaystyle \vert \int _{\Sigma }f^{*}\omega \vert \leq 4p(g-1)\pi }$.

This inequality can be interpreted as an estimate for the norm of the Kähler class (and thus the Toledo invariant ) in limited cohomology. It generalizes the estimate of the Euler class for . ${\displaystyle G/K=SL(2,\mathbb {R} )/SO(2)}$

literature

• J. Milnor: On the existence of a connection of curvature zero , Comm. Math. Helv. 21, 215-223 (1958)
• J. Wood: Bundles with totally disconnected structure group , Comm. Math. Helv. 46, 257-273 (1971)
• W. Goldman: Topological components of spaces of representations , Invent. Math. 93: 557-607 (1988)
• M. Burger, A. Iozzi, A. Wienhard: Surface group representations with maximal Toledo invariant , Ann. Math. 172, 517-566 (2010)
• T. Hartnick, A. Ott: Milnor-Wood type inequalities for Higgs bundles, online (PDF)