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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 23rd 2014
   • (edited Oct 23rd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexGiven an $n$-dimensional manifold $\Sigma$, with $B \mathrm{Diff}(\Sigma)$ the classifying space of its diffemorphism group. How different is that from the following classifying space: Write $\tau_\Sigma \colon \Sigma \to \mathbf{B} GL(n)$ for the morphism of smooth stacks that modulates the tangent bundle of $\Sigma$. Notice that an endomorphism of $\Sigma$ regarded in the slice over $\mathbf{B}\mathrm{GL}(n)$ this way is equivalently a local diffeomorphism: $$ LocalDiffeos(\Sigma, \Sigma) \simeq Smooth \infty Grpd_{/\mathbf{B}GL(n)}(\tau_\Sigma, \tau_\Sigma) \,. $$ Under geometric realization $\Pi$ the morphism $\tau_\Sigma$ becomes, equivalently $$ \Pi(\tau_\Sigma) \colon \Pi(\Sigma) \longrightarrow B O(n) \,. $$ There is then the automorphism $\infty$-group $$ Aut_{/B O(n)}(\Pi(\Sigma)) = \infty Grpd_{/B O(n)}(\tau_\Sigma, \tau_\Sigma)_{equiv} $$ of $\Pi(\Sigma)$ regarded in the slice of $\infty Grpd$ over $B O(n)$. How does $ B \mathrm{Aut}_{B O(n)}(\Pi(\Sigma))$ relate to $B \mathrm{Diff}(\Sigma)$ ? Or in other words: to which extent may we "take $\Pi$ inside" to pass from $$ \Pi( \mathrm{Aut}_{/\mathbf{B}GL(n)}(\Sigma) ) $$ to $$ \mathrm{Aut}_{/\Pi(\mathbf{B}GL(n))}(\Pi(\Sigma)) $$ ?

   Given an $n$-dimensional manifold $\Sigma$, with $B \mathrm{Diff}(\Sigma)$ the classifying space of its diffemorphism group.

   How different is that from the following classifying space:

   Write $\tau_\Sigma \colon \Sigma \to \mathbf{B} GL(n)$ for the morphism of smooth stacks that modulates the tangent bundle of $\Sigma$.

   Notice that an endomorphism of $\Sigma$ regarded in the slice over $\mathbf{B}\mathrm{GL}(n)$ this way is equivalently a local diffeomorphism:

   $$LocalDiffeos(\Sigma, \Sigma) \simeq Smooth \infty Grpd_{/\mathbf{B}GL(n)}(\tau_\Sigma, \tau_\Sigma) \,.$$

   Under geometric realization $\Pi$ the morphism $\tau_\Sigma$ becomes, equivalently

   $$\Pi(\tau_\Sigma) \colon \Pi(\Sigma) \longrightarrow B O(n) \,.$$

   There is then the automorphism $\infty$-group

   $$Aut_{/B O(n)}(\Pi(\Sigma)) = \infty Grpd_{/B O(n)}(\tau_\Sigma, \tau_\Sigma)_{equiv}$$

   of $\Pi(\Sigma)$ regarded in the slice of $\infty Grpd$ over $B O(n)$.

   How does $B \mathrm{Aut}_{B O(n)}(\Pi(\Sigma))$ relate to $B \mathrm{Diff}(\Sigma)$ ?

   Or in other words: to which extent may we “take $\Pi$ inside” to pass from

   $$\Pi( \mathrm{Aut}_{/\mathbf{B}GL(n)}(\Sigma) )$$

   to

   $$\mathrm{Aut}_{/\Pi(\mathbf{B}GL(n))}(\Pi(\Sigma))$$

   ?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 23rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave made that an [MO question](http://mathoverflow.net/q/185217/381)

   have made that an MO question