6,558 reputation
11421
bio website bangor.ac.uk/r.brown
location University of Wales-Bangor, United Kingdom
age 80
visits member for 2 years, 9 months
seen 3 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!


Feb
9
revised $2$-Morphisms in the Fundamental $2$-Groupoid
added info on a consequence
Feb
9
comment Associativity in the fundamental groupoid of a space
If you use paths as maps $f:[0,r]\to X$ where $r \geqslant 0$ then composition of paths is strictly associative and has strict identities, i.e. forms a category. See the book "Topology and Groupoids" (first edition, 1968).
Feb
8
comment How to visualize projective plane
Have a look at pages.bangor.ac.uk/~mas010/outofline/motion.html#motion for a sequence of stills showing how the projective plane represents rotations in 3-space.
Feb
7
comment Nice examples of groups which are not obviously groups
There is a lot to be said for using Moore paths in defining the fundamental group(oid). A Moore path in $X$ is a pair $(f,r)$ such that $f:[0, \infty) \to X$, $r \in [0, \infty)$, and $f$ is constant on $[r, \infty)$. The composition $(f,r) *(g,s)$ is defined if and only if $f(r)=g(0)$ and is of the form $(h,r+s)$. Composition then gives a category of Moore paths on $X$.
Feb
6
comment non homology based polynomial time computable invariants
Or restrict the applications to those $K$ for which the fundamental groupoid has this property, of which there may also be higher dimensional versions.
Feb
6
comment non homology based polynomial time computable invariants
@rhl: Do you regard the fundamental groupoid $\pi_1(K,K^0)$ of a simplicial complex $K$ as a polynomial time computable invariant? If not, then you might have to restrict invariants nearer to homotopy to those for which their fundamental groupoids are of that type. But this is just a guess.
Feb
4
revised non homology based polynomial time computable invariants
added a bit more
Feb
4
revised non homology based polynomial time computable invariants
additional comments
Feb
4
comment non homology based polynomial time computable invariants
@Mariano: Thanks for the question, as a web search has just shown me an English translation at matematicas.unex.es/~navarro/res/esquisseeng.pdf
Feb
4
answered non homology based polynomial time computable invariants
Feb
2
comment $2$-Morphisms in the Fundamental $2$-Groupoid
You meet try working with Moore rectangles: see arXiv:0909.2212 Moore hyperrectangles on a space form a strict cubical omega-category. But in the context of spaces, and no extra structure, there will always be problems with getting a strict 2-groupoid or double groupoid, in my view.
Feb
2
comment What does “Arrows are more important than objects” really mean?
I agree with Martin, and would just like to add the importance of proving a particular construction is a limit or colimit by verifying the universal property, instead of checking that a particular construction works. Another virtue of the "arrows" approach is the analogies it gives between different categories: the constructions of, say, colimits may be quite different but they are still colimits. All this contributes to the unity of mathematics.
Jan
31
revised $2$-Morphisms in the Fundamental $2$-Groupoid
added more detail
Jan
31
answered $2$-Morphisms in the Fundamental $2$-Groupoid
Jan
27
revised Alternative “functorial” proof of Nielsen-Schreier?
added a reference to mathoverflow
Jan
22
revised The fundamental group of Cayley graph
additional comments
Jan
22
answered The fundamental group of Cayley graph
Jan
21
answered Mayer-Vietoris Type Sequence For Pushouts
Jan
19
comment Fast paced book in point-set topology to move on to algebraic topology
I will take this opportunity to advertise my own book "Topology and Groupoids" since it marries the geometric and categorical viewpoints, has lots of exercises and pictures; it covers some aspects, e.g. the use of groupoids for covering spaces and orbit spaces, not dealt with elsewhere. The e-version is available via kagi at £5. This new edition is published privately to cut the cost. Available from amazon.
Jan
14
comment Properties of $\pi_n$ from a category theoretical point of view
@Zhen: The following paper will I hope explain more what I mean: 74. ``Computing homotopy types using crossed $n$-cubes of groups'', {\em Adams Memorial Symposium on Algebraic Topology}, Vol 1, edited N. Ray and G Walker, Cambridge University Press, 1992, 187-210. math.AT/0109091 . The higher homotopy groupoids I use are all in some sense cubical and are all defined using homotopy classes of certain maps; the proofs that the compositions are well defined are non trivial. So I think we do have good and useful algebraic models, but they are not easy.