Group Homology very basic question on definitions
- From: James <james545@xxxxxxxxx>
- Date: Tue, 31 Jul 2007 12:30:07 EDT
Dear all,
I have been going crazy searching for a definition for
a couple hours and haven't been able to find it on the
internets or in several books. I would be very grateful
if you could assist me with my few questions.
I am reading a paper that involves homology of groups,
and considers the 0-th, first, and 2nd homology groups
of the form H_i(G,M), where G is a group and M is a
G-module, i = 0, 1, 2. But when he talks about 1-cycles
or 2-cycles, he describes them as maps from G to M. The
only definition I have seen of homology groups is that
the n-th chain group is generated by symbols of the form
m (x) [g_1 | g_2 | ... | g_n] or something like that
(I don't have my books on me), not that they are maps
from G to M.
How do you describe the chain groups as maps from G to M
and what is the boundary operator? I would be very very
happy if you could tell me the definitions for the chain
gruops in dimension 0,1,2, as well as their boundary
maps (or a general formula, or what it means for the
chains to be cycles), or if you could provide a
reference that talks about this.
For example, in COhomology, 1-cocycles are maps
f : G ----> M such that f(ab) = f(a) a*f(b). How are
1-cycles described as maps?
Also, one last question. If we can describe 1-cycles as
maps from G to M, then if M is a G-module and M is an
H-module and there is a homomorphism G ----> H, then we
can probably get a restriction map
H_i(H,M) ----> H_i(G,M), right? I would need to check
this after I knew the definition of 1-cycles
(or n-cycles) as maps from G to M.
Thank you very very much for your help,
James
.
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