Topology, topological phases and the Nobel Prize in Physics 2016

I do not think I have deppen enough into the winning theme of this year's Nobel Prize in Physics.
The winners have been Thouless, Haldane and Kosterlitz "for theoretical discoveries of topological phase transitions and topological phases of matter".
But what does topology mean?
To begin with, the definition of topology is as follows: the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects.
You can access below an explanatory video about the topology, as a mathematical concept.
Now, geometry and mathematics, what is the physical relationship with matter?
By clicking on the image below you can access all kind of relevant information about topological phases.
Thanks to Satoshi Tanda, from Hokkaido University, for the image.