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David Roberts ♦
David Roberts ♦
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For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:

Lift diagram here.Lift diagram here: the map from M to BO(n) has an up-to-homotopy lift through the projection BSO(n) to BO(n)

There are two orientations on $M$. Is it ever true that they correspond to two different witnesses $H$ in the above diagram?

This question is motivated by the observation that an orientation on $M$ is sometimes considered an example of an "$SO(n)$-structure" on $M$, a special case of a $G$-structure on $M$, part of the data of which one takes the 2-morphism witnessing the lift of $\tau_M$ to $BG$.

For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:

Lift diagram here.

There are two orientations on $M$. Is it ever true that they correspond to two different witnesses $H$ in the above diagram?

This question is motivated by the observation that an orientation on $M$ is sometimes considered an example of an "$SO(n)$-structure" on $M$, a special case of a $G$-structure on $M$, part of the data of which one takes the 2-morphism witnessing the lift of $\tau_M$ to $BG$.

For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:

Lift diagram here: the map from M to BO(n) has an up-to-homotopy lift through the projection BSO(n) to BO(n)

There are two orientations on $M$. Is it ever true that they correspond to two different witnesses $H$ in the above diagram?

This question is motivated by the observation that an orientation on $M$ is sometimes considered an example of an "$SO(n)$-structure" on $M$, a special case of a $G$-structure on $M$, part of the data of which one takes the 2-morphism witnessing the lift of $\tau_M$ to $BG$.

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Arnav Das
Arnav Das
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Do the two orientations on an orientable manifold $M$ witness the same lifts of $\tau_M: M \to B\text{O($n$)}$ to $B\text{SO($n$)}$?

For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort:

Lift diagram here.

There are two orientations on $M$. Is it ever true that they correspond to two different witnesses $H$ in the above diagram?

This question is motivated by the observation that an orientation on $M$ is sometimes considered an example of an "$SO(n)$-structure" on $M$, a special case of a $G$-structure on $M$, part of the data of which one takes the 2-morphism witnessing the lift of $\tau_M$ to $BG$.