Connecting Homomorphism
The homomorphism 
which, according to the snake lemma, permits construction of an exact sequence
 |
(1)
|
from the above commutative diagram with exact rows. The homomorphism 
is defined by
 |
(2)
|
for all 
, 
denotes the image, and 
is obtained through the following construction, based on diagram chasing.
1. Exploit the surjectivity of 
to find 
such that 
.
2. Since 
because of the commutativity of the right square, 
belongs to 
, which is equal to 
due to the exactness of the lower row at 
. This allows us to find 
such that 
.
While the elements 
and 
are not uniquely determined, the coset 
is, as can be proven by using more diagram chasing. In particular, if 
and 
are other elements fulfilling the requirements of steps (1) and (2), then 
and 
, and
 |
(3)
|
hence 
because of the exactness of the upper row at 
. Let 
be such that
 |
(4)
|
Then
 |
(5)
|
because the left square is commutative. Since 
is injective, it follows that
 |
(6)
|
and so
 |
(7)
|
REFERENCES:
Bourbaki, N. "Le diagramme du serpent." §1.2 in Algèbre. Chap. 10, Algèbre Homologique. Paris, France: Masson, 3-7, 1980.
Lang, S. Algebra, rev. 3rd ed. New York: Springer Verlag, pp. 158-159, 2002.
Mac Lane, S. Categories for the Working Mathematician. New York: Springer Verlag, pp. 202-204, 1971.
Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub.,p. 141, 1993.
