History behind Exact Sequences. I am very much interested in listening to the history behind the exact sequence. We know that the exact sequence is sequence of objects with morphisms such that image of one morphism equals to the kernel of the next one. 
But how did the whole idea start ? . 
What is the motivation behind considering the image and kernel equality and linking groups ? . 
How did the exact sequences come into play ?. 
I want to hear some on some of the above things. 
Thank you.
 A: In A History of Algebraic and Differential Topology, Chapter 5, §5,
Dieudonné says that Hurewicz introduced exact sequences in 1941.
On the other hand, Weibel in his History of Homological Algebra [1] says that
Kelley and Pitcher coined the term “exact sequence” in 1947 [2].
[1] History of Homological Algebra, in History of topology (edited by I. M. James), 1999.
[2] Exact homomorphism sequences in homology theory, Annals
of Math. 48 (1947), 682–709.
A: The following is an extract of S. Krantz's wonderful Mathematical Apocrypha Redux which should answer your question:

It seems that the concept of exact sequence originates with Hurewicz (Bulletin of the American Mathematical Society 47(1941),562). But he did not use the term. In fact the terminology made its debut in a paper of John L. Kelley and Everett Pitcher (Annals of Mathematics 47(1947), 682-709). Although they may have the first citation in print, they humbly attribute the terminology to Eilenberg and Steenrod, who cooked up the language for use in their classic text Foundations of Algebraic Topology. Evidently, in their original draft for that book, they left a blank space for every occurrence of the idea. They were seeking just the right language, and were waiting for inspiration to hit. Eilenberg used the term "exact sequence" in a course he taught at the University of Michigan in 1946. He and Steenrod adopted it for their book, which was published in 1952. In the abstracts of talks that Kelley and Pitcher submitted in 1945 and 1946, they alluded to the idea of exact sequence, but they used the language "natural homomorphism sequence." We can be grateful that, for their published work, they adopted the more elegant language "exact sequence." That is the 
  argot that lives on today.

