quasar987 said:In Lee's Intro to topological manifolds, p.105, it is written that every manifold of dimension 3 or below is triangulable. But in dimension 4, threre are known examples of non triangulable manifolds. In dimensions greater than four, the answer is unknown.
But in Bott-Tu p.190, it is written that every manifold admits a triangulation.
Which is right?
quasar987 said:In Lee's Intro to topological manifolds, p.105, it is written that every manifold of dimension 3 or below is triangulable. But in dimension 4, threre are known examples of non triangulable manifolds. In dimensions greater than four, the answer is unknown.
But in Bott-Tu p.190, it is written that every manifold admits a triangulation.
Which is right?
lavinia said:I would guess that Bott and Tu mean every smooth manifold since their book is about differential topology. It is a theorem of Whitehead, I believe, that every smooth manifold has a smooth triangulation.
A manifold is a mathematical object that is locally similar to Euclidean space. It is a topological space that is smooth and can be described by coordinates and equations.
A manifold is triangulable if it can be divided into smaller pieces, called simplices, that can be pieced together to form the manifold. This process is similar to dividing a surface into smaller triangles.
The answer is no. There are certain manifolds, such as non-orientable manifolds, that cannot be triangulated. This is known as the Hauptvermutung, a conjecture in topology that has not yet been proven.
Triangulability of manifolds is important because it allows for easier visualization and understanding of these mathematical objects. It also has applications in fields such as differential geometry and topology.
Yes, there are certain conditions, such as the PL condition and the piecewise linear homotopy condition, that guarantee a manifold to be triangulable. However, these conditions are not easily verified for all manifolds.