Having problem with tom Dieck's algebraic topology text

Question

(An online PDF of the text Algebraic Topology by Tammo tom Dieck can be found here.)

This question is really soft. I'm having problem reading this text. Let me elaborate.

I found this book too formal at some places. I'm perfectly fine with formality if it makes things elegant, e.g., he states the Seifert–van Kampen theorem using pushouts of groupoids, which I prefer over, say, Hatcher's formulation. However, I get lost in formality if it does not lead to any "real" results. That's exactly the problem with Chapter 4 and 5, on homotopy theory (mapping cylinder, suspension, loop space, Pupper sequences, fibration and cofibration). He defines things and proves propositions that look completely technical, like "abstract nonsense".

As an example, for cofibrations he defines the transport functor (page 107, section 5.2). It is a functor $\Pi(K,X)\to\mathsf{SET}$, where $i:K\to A$ is a cofibration, object map being $(f:K\to X)\mapsto[(A,i),(X,f)]^K$, and morphism map defined thus: given a homotopy $\varphi:K\times I\to X$, we define a map $[(A,i),(X,\varphi_0)]^K\to[(A,i),(X,\varphi_1)]^K, f\mapsto\Phi_1$, where we lifting $\varphi$ to $\Phi:A\times I\to X$ with initial condition $f$. I can follow the logic, but I don't know why we care about such a functor. The author uses this to prove, among other things, that a homotopy equivalence between cofibrations is actually a cofiber homotopy equivalence (a result that can also be found in Peter May's notes). However, what is this result trying to say? Why should we prove something like that?

To be specific, I'd like to ask these questions:

1. Is every result in tom Dieck really something to be used later?
2. Could you please give briefly introduce to me what this abstract and formal part of homotopy theory is doing? And what are some interesting things we can prove with the aid of these formal results?
3. If I want to "really" understand these formal statements, what should I do? (Now I just see formal statements and formal arguments, not knowing why we care about these...)

Thanks for any help!

Edit: I have to say that I love tom Dieck's style! It's much better than Hatcher, when everything's so elegantly formulated using, e.g., categorical notions. It's just that if some results in tom Dieck are (1) too formal; (2) not found in other books; (3) have no "real" geometric meaning; (4) never used later, then why does the author include it?

Answer 1

Here are my answers to some of your questions. In the mid 1960's I was asking myself how results in the literature on spaces $X$ with well pointed base point $a$ generalised to the case where $a$ was replaced by a closed subspace $A$ such that the inclusion $i: A \to X$ was a cofibration. In particular, I looked at the operation of the fundamental group on homotopy groups (which Henry Whitehead had remarked in my hearing was a fascination to early workers in homotopy theory) and the result that a homotopy equivalence of spaces induces an isomorphism of homotopy groups. Here the space with well pointed base point is the sphere $(S^n,1)$. To my surprise, the generalisation led to a gluing theorem for homotopy equivalences, which appeared in the 1968 and subsequent editions of the book which is now Topology and Groupoids, see Sections 7.4, 7.5.

All editions of that book also treat an idea inspired by work of P.J.Higgins on the algebra of groupoids, namely the fundamental groupoid on a set $A$ of base points, which is discussed in answers to this mathoverflow question. This enables one to get away from any implicit assumption that all spaces of interest in topology are pathconnected.

December 12, 2018 To add to the first paragraph: one of the points of this "formal homotopy theory" is to give methods of constructing homotopies and homotopy equivalences, and which are systematic and not ad hoc. It is reasonable for such results to come before giving methods for, say, showing there is not a homotopy equivalence, except for say connectivity arguments.

The paper by N E Steenrod "Cohomology Operations, and Obstructions to Extending Continuous Functions*, ADVANCES IN MATHEMATICS 8, 371-416 (1972), is a good introduction to some basic problems of algebraic topology.