Classification of 4D manifolds

Higher-dimensional geometry (previously "Polyshapes").

Classification of 4D manifolds

Postby steelpillow » Sat Apr 13, 2024 6:54 pm

A standard method of analysing manifolds (n-surfaces or spaces) is to break them down into (i.e. tile them with) n-simplices and then study the "chains" of simplices thus created. I believe this approach is known as homology.

The classification of manifolds in this way has been accomplished in every dimension except 4. Some years ago I came across a result claiming that a handful of 4-manifolds remain unclassified, because they have some kind of inherent anomaly which is not definable in terms of its homology.

Can anybody point me towards some understanding of this?
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