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Jul
28
revised Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?
added a link to mathoverflow
Jul
26
comment Show $X$ is simply-connected given properties of two subsets
See also mathoverflow.net/questions/175923/…
Jul
24
comment Definition of neighborhood and open set in topology
I have added the "non empty" condition.
Jul
24
revised Definition of neighborhood and open set in topology
corrected definition and added a link
Jul
24
answered On defining homology groups
Jul
20
revised Computation of fundamental group of pseudo circle
changed "fundamental group" to the more accurate "fundamental groups"
Jul
19
revised Computation of fundamental group of pseudo circle
comment on higher homotopy groups
Jul
19
revised Computation of fundamental group of pseudo circle
added more links
Jul
18
answered Pullbacks and homotopy equivalences
Jul
9
comment What are some of the major open problems in category theory?
I have been asked about famous problems in category theory a while ago and gave some discussion on this at pages.bangor.ac.uk/~mas010/famousproblems.html . At the Aveiro CT2015 dinner I could not resist saying it was exhilarating to see in the talks, quoting the bard, "as imagination bodies forth the forms of things unknown". It seems to me the clearly formulated problems come after that process, which may be stimulated by other problems.
Jul
9
awarded  algebraic-topology
Jul
4
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?