Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added to moduli space of curves a paragraph mentioning the result by Harer-Zagier on the orbifold Euler characteristic of ℳg,1 being ζ(1−2g).
added a brief remark along these lines also to moduli space of elliptic curves – Properties – Euler characteristic
ah, so I only now realize that the Riemann moduli space is also equivalently that of almost complex structures mod orientation preserving diffeos (reviewed e.g. in Madsen 07, section 1.1).
This has one nice consequence: this means that the moduli space may be constructed in a general abstract homotopy-type theoretic way just as is discussed in some detail at general covariance.
I have added a remark on this to the entry:
We indicate how this definition has a formulation in the homotopy-type theory H of smooth homotopy types.
By the discussion at almost complex structure (see this remark), if the tangent bundle of a 2n-dimensional smooth manifold is modulated by a map
τX:X⟶BGL(2n,ℝ)to the delooping in smooth stacks of the general linear group, then an almost complex structure on X is equivalently a lift J in
XJ⟶BGL(n,ℂ)τ↘↙almCompBGL(2n,ℝ).This in turn is equivalently a map
J:τX⟶almCompin the slice (∞,1)-topos H/BGL(2n,ℝ), hence with the canonical empedding
τ(−):SmthMfdet2n↪H/BGL(2n,ℝ)understood, then almComp∈H/BGL(2n,ℝ) is the universal moduli stack of almost complex structures.
Now τX carries a canonical ∞-action by the diffeomorphism group. Using this one may canonically form the homotopy quotient
[τX,almComp]//Diff(X)by a general abstract construction that is discussed in some detail at general covariant – Formalization in homotopy type theory. For n=1 this is hence the Riemann moduli space.
Somewhat related new entry, Outer space.
1 to 4 of 4