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18h
comment Associativity of Operation * on Path-homotopy Classes Proof (Supposedly Trivial Question)
Most topology texts insist that a path in $X$ has to be a map $[0,1] \to X$. However the classic book on Knot Theory by Crowell and Fox defines a path in $X$ to be a pair $(f,r)$ where $r \geqslant 0$ and $f: [0, \infty ) \to X$ is a map constant on $[r,\infty)$. With an easily defined composition the paths in $X$ form a category, in particular associativity holds. There is a variation on this method in my book "Topology and Groupoids". These approached make it easier for students and seem more understandable.
Jul
1
comment van Kampen theorem for fundamental groupoid of $X$ relative to $A$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
Jun
25
comment Complete and unabridged proof of the theorem of acyclic models
Re free online: actually T&G is available at £5 from kagi.com, but I am aware of free downloadable versions. The main thing is I want people to read the books. The big EMS book is available on my web site courtesy of the contract with the EMS.I hope readers will also insist that both books are available in print in their institution's library! Books are a good random access device, and more readable, though non clickable, but can be heavy to carry around.
Jun
24
answered Complete and unabridged proof of the theorem of acyclic models
Jun
21
comment Applications of algebraic topology?
To add to my last comment, and to Alan's points, some central problems in mathematics and science come under the term "local-to-global". Also how can we understand the structure of higher dimensional spaces, and how this influences local-to-global problems. All this is under development, despite great progress.
Jun
21
comment Applications of algebraic topology?
@Alan U. Kennington I would not say "the tools" but "some tools". Tools are still being developed, for example in "higher dimensional algebra", see presentations on my preprint page.
Jun
21
comment Applications of algebraic topology?
@Metatheton Lack of "context" is in my view a problem with much University maths teaching and texts. There are articles on this on my "Popularisation and Teaching" web page: pages.bangor.ac.uk/~mas010/publar.html
Jun
20
comment Applications of algebraic topology?
For a more gentle, I think, study of cells and adjunction spaces you could try my book "Topology and Groupoids" (available from amazon). That book is also the only topology text in English to use the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points to compute the fundamental group of a circle, and of more general unions of non connected spaces.
Jun
20
answered Applications of algebraic topology?
Jun
13
comment Calculate the fundamental group of $S^1/\mathbb Z_n$
I point out that Chapter 11 of the book "Topology and Groupoids" (available from amazon) determines the fundamental groupoid of an orbit space under a properly discontinuous action, e.g. that of a finite group on a Hausdorff space. An older version of this chapter is available at arXiv:math/0212271. I can't work out more now as I am travelling tomorrow!
Jun
13
revised Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
added a clarification
Jun
13
comment Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
@Kevin Carlson It just seems a pity while doing this not to prove a theorem which does so much more, since you need to do the orthodox van Kampen theorem anyway. If you do the version for $\pi_1(X,C)$, for $C$ is a set of base points, you also should develop enough of the algebra of groupoids to exploit this result. Why not at least let students know there is such a result?
Jun
12
revised Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
added another diagram
Jun
11
revised Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
referred to some other work on nonabelian cohomology
Jun
11
revised Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
added a clarification
Jun
10
comment Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
I have to mark up this question because I sense the same irritation I had in 1965 when writing a text, that the usual van Kampen Theorem did not compute the fundamental group of the circle, which is THE basic example in algebraic topology, So one made in essence a detour to covering spaces to get this value. All unaesthetic! Then I was led to the groupoid arguments by reading a paper of Higgins. This felt good to me, and a meeting with George Mackey in 1967 showed me some wider possibilities in extending from groups to groupoids.
Jun
10
revised Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
slight clarifications in relation to the original question
Jun
10
comment Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
I'm only too well known as a groupoid groupie! Too bad!
Jun
10
revised Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?
added some links and correspondence
Jun
10
answered Can one prove that the fundamental group of the circle is $\mathbb Z$ without using covering spaces?