I am aware that at least for lower dimensions,

"smooth manifolds iff triangulable manifolds"

 

at least for dimensions below a certain critical dimensions D.

My question is that for

• For orientable manifolds, in which lower dimensions $\leq$ D, such that "smooth manifolds iff triangulable manifolds" is true, but above that dimensions > D is false? (So that "smooth manifolds may not be triangulable manifolds.")

Consider:

1. orientable SO-manifolds with SO(D) co/bordism structure.

2. orientable Spin-manifolds with Spin(D) co/bordism structure.

• For non-orientable manifolds, in which lower dimensions $\leq$ D, such that "smooth manifolds iff triangulable manifolds" is true, but above that dimensions > D is false? (So that "smooth manifolds may not be triangulable manifolds.")

Consider:

3. non-orientable O-manifolds with O(D) co/bordism structure.

What are their critical dimensions D?

In this post, we learn:

"All orientable 5-dimensional manifolds are triangulable." "In 6 dimensions, there are non-triangulable orientable manifolds."

 

Are these referred to topological manifolds? Or smooth manifolds?

However, I heard an alternative view from a geometry topologist that (maybe standard viewpoint to some of you, but I apologize for my background ignorance):

"All smooth manifolds are uniquely triangulable. No critical dimensions D constraint or orientability constraints."

p.s. Your correct answers are more important. References (the proofs) are secondary but very helpful.

Many thanks! Really appreciate your holiday time answer and inspiration!

I am aware that at least for lower dimensions,

"smooth manifolds iff triangulable manifolds"

at least for dimensions below a certain critical dimensions D.

My question is that for

• For orientable manifolds, in which lower dimensions $\leq$ D, such that "smooth manifolds iff triangulable manifolds" is true, but above that dimensions > D is false? (So that "smooth manifolds may not be triangulable manifolds.")

Consider:

1. orientable SO-manifolds with SO(D) co/bordism structure.

2. orientable Spin-manifolds with Spin(D) co/bordism structure.

• For non-orientable manifolds, in which lower dimensions $\leq$ D, such that "smooth manifolds iff triangulable manifolds" is true, but above that dimensions > D is false? (So that "smooth manifolds may not be triangulable manifolds.")

Consider:

3. non-orientable O-manifolds with O(D) co/bordism structure.

What are their critical dimensions D?

In this post, we learn:

"All orientable 5-dimensional manifolds are triangulable." "In 6 dimensions, there are non-triangulable orientable manifolds."

Are these referred to topological manifolds? Or smooth manifolds?

However, I heard an alternative view from a geometry topologist that (maybe standard viewpoint to some of you, but I apologize for my background ignorance):

"All smooth manifolds are uniquely triangulable. No critical dimensions D constraint or orientability constraints."

p.s. Your correct answers are more important. References (the proofs) are secondary but very helpful.

Many thanks! Really appreciate your holiday time answer and inspiration!