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- TL;DR Summary
- Wikipedia explanation about cohomology is very obscure to me and I'm wondering whether I can find here help to translate it in simpler terms.

Wikipedia says that a general projective representation cannot be lifted to a linear representation and the obstruction to this lifting can be understood via group cohomology.

For example, I see that a spin group is a central extension of SO(3) by Z/2.

More generally I can follow the reasoning that central extensions of Lie groups by

But while I can easily admit that the discrete central group above

For example, I see that a spin group is a central extension of SO(3) by Z/2.

More generally I can follow the reasoning that central extensions of Lie groups by

**discrete groups**are covering groups and all projective representations of G are linear representations of the universal cover, hence no central charges occur.But while I can easily admit that the discrete central group above

**happens to be**(isomorphic to) the fundamental group of the Lie group G, I can't really grasp how the homotopy and cohomology enter here**by their definitions**? How the second cohomology group is in one-to-one correspondence with the set of central extensions?