# Why did we study “defect” in the late 1800s?

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.

• Why not skip the "motivation" and study the maths? Aug 10, 2019 at 18:45
• @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... Aug 10, 2019 at 18:50
• If you want motivation, read any text on algebraic topology. Aug 10, 2019 at 18:51
• @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”... Aug 10, 2019 at 18:59
• 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. Aug 10, 2019 at 19:45