Morita Miller Mumford classes

In mathematics , the Morita-Miller-Mumford classes or MMM classes are characteristic classes of bundles of areas.

MMM classes of bundles of areas

For an area bundle , its Morita-Miller-Mumford classes are defined by ${\displaystyle E\to B}$

${\displaystyle \kappa _{n}(E)=(-1)^{n+1}\pi _{!}((c_{1}(T_{v}E))^{n+1})\in H^{2n}(B;\mathbb {Z} ),n=1,2,\ldots ,}$

where is the vertical tangent bundle , its first Chern class and the Gysin homomorphism . ${\displaystyle T^{v}E}$ ${\displaystyle c_{1}(T^{v}E)}$ ${\displaystyle \pi _{!}}$

Modular space of Riemannian surfaces

Mumford considered the modular space of Riemann surfaces with its universal bundle of surfaces and its tautological classes (the MMM classes) and made the conjecture, later named after him, that the cohomology of the modular space is a polynomial algebra in the tautological classes in sufficiently small degrees. ${\displaystyle M_{g}}$

Cohomology of mapping class groups

With rational coefficients one has an isomorphism

${\displaystyle H^{*}(M_{g};\mathbb {Q} )=H^{*}(B\Gamma _{g};\mathbb {Q} )}$

between the cohomology of the module space and the group cohomology of the mapping class group . The tautological classes of the universal area bundle thus define classes in the cohomology of the mapping class group.

From the Mumford conjecture (and Harer's stability theorem) it follows that the cohomology of the mapping class group agrees in degrees with the polynomial algebra in the MMM classes. ${\displaystyle \leq {\frac {2}{3}}(g-1)}$

literature

• David Mumford : Towards an enumerative geometry of the moduli space of curves in Arithmetic and Geometry (M. Artin and J. Tate, editors), Progr. Math., Vol. 36, Birkhäuser, pp. 271-328, 1983.
• EY Miller: The homology of the mapping class group , vol. 24, J. Diff. Geom., Pp. 1-14, 1986.
• S. Morita: Characteristic classes of surface bundles , vol. 90, Invent. Math., Pp. 551-577, 1987.