TITLE: Topology and the 2016 Nobel Prize in Physics
QUESTION [119 upvotes]: I was very happy to learn that the work which led to the award of the 2016 Nobel Prize in Physics (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In particular, the prize was awarded "for theoretical discoveries of topological phase transitions and topological phases of matter".
Having read both the popular and advanced versions of the scientific background found on the website linked above, I'm left with the impression that the most advance topological concepts used are winding numbers, vector fields and the Poincaré-Hopf index theorem.
I wonder if anyone who reads MO is sufficiently familiar with this work to explain to me why this impression is correct or incorrect. To ask a precise question:

Which topological concepts and results are involved in the work which led to the award of the 2016 Nobel Prize in Physics?

And a follow-up question:

Where should a topologist go to read about topological phases of matter and topological phase transitions?

REPLY [16 votes]: The involved topology are
1) $\pi_1(S^1)$ in TK transition  (Thouless-Kosterlitz)
2) First Chern class and Chern number in TKNN relation to Hall conductance (Thouless) and in Haldane model for quantum anomalous Hall effect (Haldane)
3) $\pi_2(S^2)$ in Haldane phase ($\theta$-topological term) (Haldane)
Now some physics. 1) TK transition is a transition between non-topological phases induced by topological defects (vortexes). Thouless and Kosterlitz revealed the importance of the topological defects in phase transition. TK transition is also the first example of a continuous transition that does not change the symmetry. 2) TKNN related the Chern number to quantized Hall conductance in integer quantum Hall effect on lattice. Haldane showed that integer quantum Hall effect can appear on a lattice if we break the time reversal and parity without the need of external magnetic field. 3) Haldane showed that spin-1/2 chain has a $\theta$-topological term with $\theta=\pi$, and spin-1 chain has a $\theta$-topological term with $\theta=2\pi \sim 0$. Thus he concluded that spin-1/2 chain is gapless (agreeing with previously known results) while spin-1 chain is gapped (this was a new result and a surprise. The gapped phase is named Haldane phase). 
Later, algebraic topology enter the study of topological phases, such as group coohomology or cobordism (classify SPT phases, including Haldane phase), unitary fusion category (classify 2+1D topological orders with gappable boundary), unitary modular tensor category (classify 2+1D topological orders), $G$-crossed unitary modular tensor category or unitary braided fusion category over a symmetry fusion category (classify 2+1D bosonic/fermionic topological orders with symmetry), etc.  Are those "quantum topology"?