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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to _[[moduli space of curves]]_ a paragraph mentioning the result by Harer-Zagier on the [orbifold Euler characteristic](http://ncatlab.org/nlab/show/moduli+space+of+curves#OrbifoldEulerCharacteristic) of $\mathcal{M}_{g,1}$ being $\zeta(1-2g)$.

   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 $\zeta(1-2g)$.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2014
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   Author: Urs
   Format: MarkdownItexadded a brief remark along these lines also to _[moduli space of elliptic curves -- Properties -- Euler characteristic](http://nlab.mathforge.org/nlab/show/moduli+stack+of+elliptic+curves#EulerCharacteristic)_

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 16th 2014
   • (edited Jul 16th 2014)
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   Author: Urs
   Format: MarkdownItexah, 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](http://ncatlab.org/nlab/show/moduli+space+of+curves#Madsen07)). 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](http://ncatlab.org/nlab/show/moduli+space+of+curves#InTermsOfTheHigherToposOfSmoothStacks) on this to the entry: *** We indicate how this definition has a formulation in the [[homotopy type theory|homotopy-type theory]] $\mathbf{H}$ of [[smooth homotopy types]]. By the discussion at _[[almost complex structure]]_ (see [this remark](complex+structure#InTermsOfSmoothStacks)), if the [[tangent bundle]] of a $2n$-dimensional [[smooth manifold]] is modulated by a map $$ \tau_X \;\colon\; X \longrightarrow \mathbf{B}GL(2n,\mathbb{R}) $$ to the [[delooping]] in [[smooth stacks]] of the [[general linear group]], then an [[almost complex structure]] on $X$ is equivalently a lift $J$ in $$ \array{ X && \stackrel{J}{\longrightarrow} && \mathbf{B} GL(n,\mathbb{C}) \\ & {}_{\mathllap{\tau}}\searrow && \swarrow_{\mathrlap{almComp}} \\ && \mathbf{B} GL(2n,\mathbb{R}) } \,. $$ 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](general+covariance#InHomotopyTypeTheory)_. For $n =1$ this is hence the Riemann moduli 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

   $$\tau_X \;\colon\; X \longrightarrow \mathbf{B}GL(2n,\mathbb{R})$$

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

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

   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.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJul 16th 2014
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   Author: zskoda
   Format: MarkdownItexSomewhat related new entry, [[Outer space]].

   Somewhat related new entry, Outer space.