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How are topological invariants constructed?

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I've seen several different definitions for what are called topological invariants, for instance in the context of

I would like to understand a bit more how they are constructed. Are they restriction of a generic mathematical theory (perhaps the famous Chern-construction), or are they constructed independently for each case (for instance, for each physical property they have, like half-charge for Majorana mode and chiral mode for 3D-TI) ?

A subsidiary question (which could eventually be postponed to an other post) would be: do these topological invariants appear in the calculation of some physical quantities ? For instance, the Chern-number appeared explicitly in quantum Hall systems into the calculation of the conductivity. What about the above examples ?

Explicit constructions would be greatly appreciate.

This post imported from StackExchange Physics at 2015-02-26 12:10 (UTC), posted by SE-user FraSchelle
asked Feb 25, 2015 in Theoretical Physics by FraSchelle (355 points) [ no revision ]

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