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I just began reading Weibel’s An introduction to Homological Algebra and stuck at the first page, where the author attempts to give a historical motivation for the homology construction:

Homological algebra is a tool used in several branches of mathematics: algebraic topology, group theory, commutative ring theory, and algebraic geometry come to mind. It arose in the late 1800s in the following manner. Let $f$ and $g$ be matrices whose product is zero. If $g\cdot v=0$ for some column vector $v$, say, of length $n$, we cannot always write $v=f\cdot u$. This failure is measured by the defect $$d=n-\text{rank}(f)-\text{rank}(g).$$ In modern language, $f$ and $g$ represent linear maps $$U\xrightarrow{f} V\xrightarrow{g} W$$ with $gf=0$, and $d$ is the dimension of the homology module$$H=\ker (g)/f(U).$$

I am not seeing any motivation for studying the “defect” of the above phenomenon. Could anyone give a brief explanation for why did the mathematicians study how to write a column vector of $f$ by product of $f$ and another column vector? Much appreciated.

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    $\begingroup$ Why not skip the "motivation" and study the maths? $\endgroup$ Aug 10, 2019 at 18:45
  • $\begingroup$ @LordSharktheUnknown Till now I don’t have any “algebraic motivation” for the construction of homology groups. I guess I might get more comfortable with homology groups if I have more motivation to study it. I could be wrong... $\endgroup$ Aug 10, 2019 at 18:50
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    $\begingroup$ If you want motivation, read any text on algebraic topology. $\endgroup$ Aug 10, 2019 at 18:51
  • $\begingroup$ @LordSharktheUnknown I have read some algebraic texts and the only motivation I gain is that the homology group “counts the holes” of a topological space. It seems more appropriate to view homology group as a functor from Top to Grp that is homotopical invariant in my opinion. When I study algebraic topology I prefer to view a construction just as a “construction”... $\endgroup$ Aug 10, 2019 at 18:59
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    $\begingroup$ Unfortunately, I don't know this subject, but you are right to ask if anyone has some insight (what you call "motivation") about the historical path that has lead in a more or less natural way to the discovery of such and such mathematical concept. New ideas cannot be traced back to older ones in only some rare cases. $\endgroup$
    – Jean Marie
    Aug 10, 2019 at 19:45

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