Hi everyone,
What is a short exact sequence?
Looking this up on Wikipedia I see that it is a "sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next".
But I'm having trouble understanding the significance of this and why it is important.
I came across it because I'm trying to understand spin group which involves the short exact sequence:
1 -> Z2 -> Spin(n) -> SO(n) -> 1
I can see that there is a 1:2 mapping between 1 and Z2 and I can see that there is a 2:1 mapping between Spin(n) and SO(n) but I can't quite see how the rest maps together?
In order to try to understand this I have tried to calculate the image and kernel for 3D rotations as represented by a unit quaternion 'q', working this out below I think I have made it fit the definition:
ker (1 -> {0,1}) = 1
im (1 -> {0,1}) = {0,1}
ker ({0,1} -> q ) = {0,1}
im ({0,1} -> q ) = {q,-q}
ker (q -> {q,-q}) = {q,-q}
im (q -> {q,-q}) = {1 + 0 i + 0 j + 0k,-1 - 0 i - 0 j - 0k}
ker ({q,-q} -> 1) = {1 + 0 i + 0 j + 0k,-1 - 0 i - 0 j - 0k}
im ({q,-q} -> 1) = 1
However I'm not sure if I have got this right and, more importantly, I can't understand the significance of exact sequences and why they are important. Can anyone shed some light on this?
thanks, Martin
P.S. I hope I have posted this in the correct forum, the subject seems to be on the border of algebra, geometry and topology.