Terribly sorry if this has been asked, but I'm not about to search 382 pages of technical questions in the field.
I am trying to develop a very basic understanding of what algebraic topology is about. Tried Hatcher, who simply dives into vague terminology I've not heard for reasons that are not clear. The heck is an $n$-cell...? What do you mean, "attached?" How in the heck are $f'$ and $f''$ from $X\to Y$ "connected" by a homotopy...?
I go into May's "Concise" book, which has at least clarified the fundamental idea of the field (algebraic invariants under continuous deformations of the topology, I gather), but the first two sentences leave me wondering how intuitive this book is actually going to be, when it comes to structures with which I'm not already familiar. And he clarifies what Hatcher meant by "connected": $h(x,0) = p(x)$ and $h(x,1) = q(x)$ "connects" $p$ and $q$ by a homotopy $h$... I can see that. But extrapolating from many answers on this site I have read, I will be judged harshly for not simply intuiting this from the vague wording. That's okay, you can think I'm stupid. I've got no problem with that.
But what I'm grasping for, and not finding anywhere, is why I should be interested in algebraic topology in the first place. What does it actually tell us about geometry? I am interested in what can be done in a space where algebraic structures is commensurable with continuous maps, but I am seriously questioning whether the field, as it is, has any true insight to offer me here. It doesn't seem like they're investigating structures with any real relevance to anything, anywhere. I am sure that I am simply lacking imagination here. I'm struggling enough to even grasp the basic concepts in any other light than pure symbol-games, let alone understand what the 'big' theorems are about. Why should I care about fundamental groups or homotopy type? What does it MEAN...? What is algebraic topology REALLY about...? What's the point of it? What does it apply to? I am sure it applies to many results in higher math, but this doesn't interest me. What disciplines in sciences and engineering have made the greatest use of results and language of algebraic topology?
I am starting to have the feeling that nobody really has any idea what they're talking about in these fields. Please prove me wrong. I am utterly perplexed as to what the point is.