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Atrium > Mathematics, Physics & Philosophy: diffeomorphisms and homotopy equivalences sliced over BO(n)

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- 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -dimensional manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>$ , with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi mathvariant="normal">Diff</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \mathrm{Diff}(\Sigma)</annotation></semantics></math>$ the classifying space of its diffemorphism group.

How different is that from the following classifying space:

Write $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub><mo lspace="0.11111em">:</mo><mi>Σ</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_\Sigma \colon \Sigma \to \mathbf{B} GL(n)</annotation></semantics></math>$ for the morphism of smooth stacks that modulates the tangent bundle of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>$ .

Notice that an endomorphism of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>$ regarded in the slice over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi mathvariant="normal">GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathrm{GL}(n)</annotation></semantics></math>$ this way is equivalently a local diffeomorphism:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>LocalDiffeos</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Smooth</mi><mn>∞</mn><msub><mi>Grpd</mi> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> LocalDiffeos(\Sigma, \Sigma) \simeq Smooth \infty Grpd_{/\mathbf{B}GL(n)}(\tau_\Sigma, \tau_\Sigma) \,. </annotation></semantics></math>$
Under geometric realization $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>$ the morphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\tau_\Sigma</annotation></semantics></math>$ becomes, equivalently
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><mo stretchy="false">)</mo><mo lspace="0.11111em">:</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(\tau_\Sigma) \colon \Pi(\Sigma) \longrightarrow B O(n) \,. </annotation></semantics></math>$
There is then the automorphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>$ -group
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>Aut</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>∞</mn><msub><mi>Grpd</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><msub><mo stretchy="false">)</mo> <mi>equiv</mi></msub></mrow><annotation encoding="application/x-tex"> Aut_{/B O(n)}(\Pi(\Sigma)) = \infty Grpd_{/B O(n)}(\tau_\Sigma, \tau_\Sigma)_{equiv} </annotation></semantics></math>$
of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(\Sigma)</annotation></semantics></math>$ regarded in the slice of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>$ over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B O(n)</annotation></semantics></math>$ .

How does $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><msub><mi mathvariant="normal">Aut</mi> <mrow><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B \mathrm{Aut}_{B O(n)}(\Pi(\Sigma))</annotation></semantics></math>$ relate to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi mathvariant="normal">Diff</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \mathrm{Diff}(\Sigma)</annotation></semantics></math>$ ?

Or in other words: to which extent may we “take $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>$ inside” to pass from
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi mathvariant="normal">Aut</mi> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Pi( \mathrm{Aut}_{/\mathbf{B}GL(n)}(\Sigma) ) </annotation></semantics></math>$
to
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="normal">Aut</mi> <mrow><mo stretchy="false">/</mo><mi>Π</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathrm{Aut}_{/\Pi(\mathbf{B}GL(n))}(\Pi(\Sigma)) </annotation></semantics></math>$
?
- 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

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Atrium > Mathematics, Physics & Philosophy: diffeomorphisms and homotopy equivalences sliced over BO(n)