About the definition of homology

Question

can someone explaine me this definition of Homology:

"The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact."

I know that homology measure the number of the holes, and the holes are given by the cycle...

Thank you.

Answer 1

If a complex is exact, its homology is zero. It follows that if its homology is non-zero, the complex is not exact and, intuitively, the larger the homology, the farther the complex is from being exact. There is nothing more than this.

Answer 2

This was going to be a comment but it does not quite fit.

It sound like your question has something to do with bridging the divide between an intuitive or "picture" understanding of homology and the formal definition of singular or simplicial homology in the language of chain complexes.

You can of course define homology without using the language of chain complexes (stable homotopy of the eilenberg-maclane spectrum smash a space is one, but there are others, including bordism theories).

It sounds like you're hovering around the issue of why should a "boundaryless object that is not itself a boundary" (i.e. homology of a chain complex) really represent holes. The short answer -- any "boundaryless" object is canonically the boundary of another object -- that other object is the cone on your original object. So one has a hole in a space $X$ if you can find a closed "subobject" $Y$ in it, so that the inclusion map $Y \to X$ does not extend to a continuous function $CY \to X$ where $CY$ is the cone.

If you think about it, this is basically what inexactness of the chain complex is measuring. Although the "cone" abstractly exists, there may be no way for it to exist in the space you are studying ($X$).

In a way I've just translated the notion of "hole" to another standard construct in algebraic topology -- the extension problem. Homotopy groups of a space are more canonical examples of extension problems describing holes.