5,095 reputation
1220
bio website bangor.ac.uk/r.brown
location University of Wales-Bangor, United Kingdom
age 79
visits member for 2 years
seen 7 hours ago

I am Professor Emeritus at Bangor University. I was an undergraduate and postgraduate at Oxford University (1953-1959), where my supervisor was the inimitable Henry Whitehead. When he died suddenly in 1960, Michael Barratt took charge, and I got my DPhil in 1962. I was lecturer at Liverpool (1959-64), Senior Lecturer and then Reader at Hull University (1964-1970), and then Professor at Bangor from 1970.

I published a text "Elements of Modern Topology" with McGraw Hill in 1968, and the 3rd edition is now available as "Topology and Groupoids" from amazon, see my web page. It was writing this book, and trying to clarify certain points, such as the fundamental group of the circle, that got me into the area of groupoids; this suggested the area of higher groupoids; research on this got going in the 1970s, and has been a major area in my work, with fortunate collaborations, and contributions from research students.

You can see most of my publications on my web site, on such topics as general topology, algebraic topology, group theory, category theory, with many on aspects of groupoids and their generalisations. There are also papers on popularisation and teaching, and on the sculptor John Robinson. One main book is the editions (1968, 1988) of the book which is now "Topology and groupoids" (2006), the last published privately to keep the price down.

I am also a joint author of a 703 page book "Nonabelian algebraic topology" published in 2011 by the European Mathematical Society. It sets out a quite new framework for basic algebraic topology, based on work over 40 years, mainly jointly with Philip Higgins, on the development and applications of higher order Seifert-van Kampen theorems, and related results. A pdf is available on my web page.

I have given a number of general lectures, to audiences from children to other scientists, including a Royal Institution Friday Evening Discourse "Out of Line" in 1992. Links to this and to various articles and presentations are available from my Preprint page and from my Popularisation and Teaching page.

See also the Popularisation of Mathematics web site http://www.popmath.org.uk for symbolic sculptures and knots!


1d
comment Spaces with fundamental group $\mathbb{Z}$
I am not sure how Lee Mosher gets the value $\mathbb Z$ without more algebra. You are right that modelling the usual proof for the circle assumes the required covering spaces exist. That is why in the 1960s I was looking for an alternative proof. Of course the groupoid result also requires the development of some algebra of groupoids, but this is generally useful.
Apr
14
revised Fundamental Group of Punctured Plane
added a link to mathoverflow
Apr
14
comment fundamental group of $\mathbb{C^*}/\{e,a\}$
Agree with the last comment! I gave a response to that question.
Apr
14
revised Fundamental Group of Punctured Plane
typo
Apr
14
answered Fundamental Group of Punctured Plane
Apr
11
revised Spaces with fundamental group $\mathbb{Z}$
added a point emphasising the little difficulty of proving the many pointed case
Apr
10
revised Spaces with fundamental group $\mathbb{Z}$
added a link to a full proof
Apr
9
revised Spaces with fundamental group $\mathbb{Z}$
improved a link
Apr
9
revised Spaces with fundamental group $\mathbb{Z}$
added an example
Apr
9
answered Spaces with fundamental group $\mathbb{Z}$
Apr
8
awarded  Yearling
Apr
6
comment Why is $\pi_1(X,x_0)$ a group?
@Jose The question is whether to use $f+g$ for addition of paths to mean first $f$ and then $g$ or to use the usual category theory convention. This is a long standing issue as to whether to write functions on the left or right of their arguments. Philip Higgins liked the algebraist's convention, functions on the right! (see his book, "Categories and Groupoids"). This leads to the rule that for an equivalence relation regarded as groupoid you have $(x,y)(y,z)=(x,z)$, which makes a lot of sense. For higher groupoids, I have found this order rather necessary.
Apr
4
comment Path-connected Space. Show abelian iff.
The question is really about an algebraic result on groupoids. If $G$ is a groupoid, and $a:x \to y$ in $G$, then there is an isomorphism $a_\#:G(x) \to G(y)$ given by conjugation by $a$. The result is that $a_\#=b_\#$ for all $a,b: x \to y$ if and only if $G(x)$ is abelian.
Apr
3
comment What does base point by us for algebraic topology?
See my answer to mathoverflow.net/questions/40945/… for looking for more freedom from the single base point view. There are also strict higher homotopy groupoids.
Mar
28
revised Is a direct limit of topological groups always a topological group?
added about convenient categories
Mar
28
answered Is a direct limit of topological groups always a topological group?
Mar
27
answered $G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian
Mar
23
revised Why is $\pi_1(X,x_0)$ a group?
added an account of another approach.
Mar
20
comment if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.
@George: It is useful to point out that restriction is a special case of pullback. In general, I don't think that beginners should be deprived of the fact that a more general point of view is possible, and let them decide. This type of comment may also be useful for more experienced readers, and writers.
Mar
20
comment Why is $*$ defined only for homotopy classes, and not individual paths between points?
It is also possible to get a groupoid algebraically by taking the category of Moore paths $(r,f)$ and adding the relations which make sure that $-(r,f)$ becomes the inverse of $(r,f)$. I don't know of any use of this construction!