Somewhat related new entry, Outer space.

]]>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 $\mathbf{H}$ 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

to the delooping in smooth stacks of the general linear group, then an almost complex structure on $X$ is equivalently a lift $J$ in

$\array{ X && \stackrel{J}{\longrightarrow} && \mathbf{B} GL(n,\mathbb{C}) \\ & {}_{\mathllap{\tau}}\searrow && \swarrow_{\mathrlap{almComp}} \\ && \mathbf{B} GL(2n,\mathbb{R}) } \,.$This in turn is equivalently a map

$J \;\colon\; \tau_X \longrightarrow almComp$in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$, hence with the canonical empedding

$\tau_{(-)} \;\colon\; SmthMfd_{2n}^{et} \hookrightarrow \mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$understood, then $almComp \in \mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$ is the universal moduli stack of almost complex structures.

Now $\tau_X$ carries a canonical ∞-action by the diffeomorphism group. Using this one may canonically form the homotopy quotient

$[\tau_X, almComp]//Diff(X)$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.

added a brief remark along these lines also to *moduli space of elliptic curves – Properties – Euler characteristic*

added to *moduli space of curves* a paragraph mentioning the result by Harer-Zagier on the orbifold Euler characteristic of $\mathcal{M}_{g,1}$ being $\zeta(1-2g)$.