Madsen and Weiss theorem

In mathematics, the Mumford conjecture or theorem of Madsen and Weiss is a theorem about the cohomology of the mapping class group or the modular space of Riemann surfaces .

The proof comes from Ib Madsen and Michael Weiss .

statement

Let the compact , orientable surface be of the gender with edge components, and be its mapping class groups , the representatives of which define all edge components. ${\ displaystyle \ Sigma _ {g, b}}$ ${\ displaystyle g}$ ${\ displaystyle b}$ ${\ displaystyle \ Gamma _ {g, b}}$

For is ${\ displaystyle b> 0}$

{\ displaystyle \ Phi \ colon \ Gamma _ {g, b} \ to \ Gamma _ {g + 1, b}}

defined by the fact that the representatives are continued through the identity map on the additional handle.

The stability set of Harer says that an isomorphism in group cohomology in degrees induced and that the cohomology groups in this area regardless of are, so you to can be limited. ${\ displaystyle \ Phi}$ ${\ displaystyle * \ leq {\ frac {2} {3}} (g-1)}$ ${\ displaystyle b> 0}$ ${\ displaystyle b = 1}$

The stable cohomology of the mapping class group can thus be defined as being sufficiently large . The stable cohomology is noted as . ${\ displaystyle H ^ {*} (B \ Gamma _ {g, 1})}$ ${\ displaystyle g}$ ${\ displaystyle H ^ {*} (B \ Gamma _ {\ infty})}$

The Mumford Conjecture said that

{\ displaystyle H ^ {*} (B \ Gamma _ {\ infty}; \ mathbb {Q}) = \ mathbb {Q} \ left [\ kappa _ {1}, \ kappa _ {2}, \ ldots \ right]}

with the Morita Miller Mumford classes . (Mumford formulated this assumption for the cohomology of the modular space of Riemann surfaces, which for rational coefficients, however, agrees with the cohomology of the mapping class group.) ${\ displaystyle \ kappa _ {i} \ in H ^ {2i} (B \ Gamma _ {\ infty}; \ mathbb {Q})}$

Madsen-Weiss proved that there is a homotopy equivalence

{\ displaystyle \ mathbb {Z} \ times B \ Gamma _ {\ infty} ^ {+} \ to \ Omega ^ {\ infty} \ mathbb {C} P _ {- 1} ^ {\ infty}}

Has. The Mumford conjecture follows in particular.

Generalization in a higher dimension

For is the algebra generated by the generalized Morita-Miller-Mumford classes , where all monomials of degree greater than pass through in which for does not occur. ${\ displaystyle W_ {g} = \ sharp ^ {g} S ^ {n} \ times S ^ {n}}$ ${\ displaystyle \ lim _ {g \ to \ infty} H ^ {*} (B \ mathrm {Diff} (W_ {g}, D ^ {2n}); \ mathbb {Q})}$ ${\ displaystyle \ kappa _ {c}}$ ${\ displaystyle c \ in H ^ {*} (B \ mathrm {SO} (2n); \ mathbb {Q}) = \ mathbb {Q} \ left [p_ {1}, \ ldots, p_ {n-1 }, e \ right]}$ ${\ displaystyle 2n}$ ${\ displaystyle p_ {i}}$ ${\ displaystyle i <{\ frac {n} {4}}}$

literature

G. Powell: The Mumford conjecture (after Madsen and Weiss) . Séminaire Bourbaki. Vol. 2004/2005. Astérisque 307 (2006), Exp. 944, 247-282
I. Madsen, M. Weiss: The stable moduli space of Riemann surfaces: Mumford's conjecture. Ann. of Math. (2) 165 (2007), no. 3, 843-941.
S. Galatius, I. Madsen, U. Tillmann, M. Weiss: The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195-239.
Y. Eliashberg, S. Galatius, N. Mishachev: Madsen-Weiss for geometrically minded topologists. Geom. Topol. 15 (2011), no. 1, 411-472.

Web links

A. Hatcher: A short exposure of the Madsen-Weiss theorem
N. Wahl: The Mumford conjecture, Madsen-Weiss and homological stability for mapping class groups of surfaces
S. Galatius: Lectures on the Madsen-Weiss Theorem

Individual evidence

^ S. Galatius, O. Randal-Williams: Stable moduli spaces of high-dimensional manifolds. Acta Math. 212 (2014), no. 2, 257-377.

[1] S. Galatius, O. Randal-Williams: Stable moduli spaces of high-dimensional manifolds. Acta Math. 212 (2014), no. 2, 257-377.