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20h
comment Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map?
I refer to this mathoverflow discussion mathoverflow.net/questions/40945/…, and the picture in my answer. The point is that the result asked for is quite subtle, but should be seen as part of the more general situation of gluing non path connected spaces. A tool here is the notion of the fundamental groupoid on a set of base points.
2d
comment First book on algebraic topology
@N.H. With this approach, the paths with composition form a category, with associativity, etc. Also it is more intuitive: if you think of a path as a journey, we are used to journeys of different length, and their composition. This avoids some proofs students find tricky, and technical, and which do not really convey much. Of course you have at some stage to discuss reparametrization, and the definition of equivalence of paths.
2d
comment First book on algebraic topology
I ought to give the url pages.bangor.ac.uk/~mas010/topgpds.html, which has a link to a £5 e-version. For more on groupoids, see the free download tac.mta.ca/tac/reprints/articles/7/tr7abs.html, ad how to fit this with homology theory see pages.bangor.ac.uk/~mas010/nonab-a-t.html, with a free pdf, and which discusses history, intuitions, and motivation.
2d
comment First book on algebraic topology
See this mathoverflow discussion mathoverflow.net/questions/40945/… for some of the issues behind my own book "Topology and Groupoids", (T&G), issues ignored by almost all other texts. T&G gives good motivation, assumes only some knowledge of analysis, has lots of figures, and exercises, does not do homology, introduces category theory, and includes topics such as orbit groupoids not even ventured into by other texts. It allows paths as maps $[0,r] \to X$, not just $r=1$.
May
18
answered $S^m * S^n \approx S^{m+n+1}$
May
14
revised intuition on the fundamental group of $S^1$
added 873 characters in body
May
14
revised Pull-back of a fibration along a homotopy equivalence
give further explanation
May
14
answered intuition on the fundamental group of $S^1$
May
13
comment Higher homology groups of infinite cyclic cover
Since you mention direct limits (=colimits) and actions, and ask to share ideas, you could look at the book advertised at pages.bangor.ac.uk/~mas010/nonab-a-t.html, particularly Chapter 8, e.g. the example of $\pi_n$ of the universal cover of $S^n\vee S^1\vee S^1$ in the Introduction to that chapter. I'd like to see these ideas applied to knots!
May
3
answered Pull-back of a fibration along a homotopy equivalence
May
2
revised Visualizing products of $CW$ complexes
typo
May
2
revised Visualizing products of $CW$ complexes
added an account of a pyjama trick
May
2
revised Visualizing products of $CW$ complexes
slight clarification and a correction to 1/2 at the end
May
2
answered Visualizing products of $CW$ complexes
May
1
comment How to compute a homotopy to show the operation on the fundamental group is assoicative?
This is an interesting exercise in linearity, but I have not heard it explained why so many (most?) topology texts insist that paths have to be "of length 1", i.e. maps $f:[0,1]\to X$, rather than of length $r \geqslant 0$, i,e, maps $[0,r] \to X$. With the more general definition, the paths in a space form a category under composition. You still have to do something on reparametrisation, and a notion of equivalence. Almost all texts also avoid the fundamental groupoid on a set of base points,.
Apr
30
revised An equivalence of categories
added the worh"uniquely", and a modification
Apr
30
answered An equivalence of categories
Apr
30
answered The Plate Trick and $SO(3)$
Apr
27
answered relative homotopy groups
Apr
26
revised Prove that exist bijection between inverse image of covering space
added another link