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(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?

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  • $\begingroup$ Further explanation of notation: $\Pi(K,X)$ is a groupoid: the objects are maps $f:K\to X$ and morphisms $f\to g$ are equivalence classes of homotopies $f\simeq g$, where two homotopies $H,H':K\times I\to X$ between $f$ and $g$ are equivalent if they are homotopic relative to $K\times\partial I$. $\endgroup$ – Colescu Dec 6 '18 at 3:14
  • $\begingroup$ I think you're asking too much. tom Dieck's book is just one, fairly formal, approach to homotopy theory. If you feel it doesn't suit you, then consider using a different text. I imagine that it is written the way it is because Hatcher's book already existed - it didn't make sense to just rewrite the same book. $\endgroup$ – Tyrone Dec 6 '18 at 9:41
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    $\begingroup$ FWIW, I hated Hatcher, and could never understand the less formal presentation of the material in his book, much preferring tom Dieck's concise statements. I just don't see what Hatcher is trying to say most of the time. For me, a functor is a concrete object I can understand, and I just get lost in Hatcher's flowing prose. The point is that I would advocate just following your mathematical muse: if you like one book over another, read your favourite. If your opinion changes later on, read the other. Just be flexible and keep moving mathematically forwards. $\endgroup$ – Tyrone Dec 6 '18 at 9:48
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    $\begingroup$ I love Hatcher and find it slow going to follow what tom Dieck is saying sometimes because it is overly formal for me, which I have to translate into something I understand better. That said, I can only agree wholeheartedly with Tyrone's comment: your goal is to find the way you learn best. $\endgroup$ – user98602 Dec 6 '18 at 12:39
  • $\begingroup$ @Tyrone Quite the contrary, I love tom Dieck's style. 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? $\endgroup$ – Colescu Dec 6 '18 at 13:25
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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.

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