Four more on Van Kampen Thm..........DMD __________________________________________________________ Subject: Re: four responses From: "Ronald Brown" Date: Tue, 8 Feb 2005 22:27:00 -0000 reply to r.brown@bangor.ac.uk Here is some more on the van Kampen theorem. 1) Philip Higgins' 1971 book has been recently reissued as a TAC (Theory and Applications of Categories) Reprint., so is downloadable. The generalisation of Grusko's theorem should be noted. 2) The motivation of my use of groupoids in my book (1968, 1988) was derived from Higgins' use of free products with amalgamation of groupoids (1963). I especially liked his use also of the groupoid obtained from G by identification of vertices, which I now think of as an induced construction f_*(G) determined by a map f: Ob(G) \to X, left adjoint to pullback f^*. 3) Some of the combinatorial groupoid theory was further developed in the second 1988 edition, including a derivation of a formula in van Kampen's paper for the non connected intersection case. It is important to consider the fundamental groupoid *on a set of base points*, chosen according to the geometry of the situation. That edition also had an improved account of covering space theory, and an account of the fundamental groupoid of an orbit space by a discontinuous action (now reissued as a Bangor preprint). 4) It seemed to me that 1-dimensional homotopy theory was phrased in a prettier way using groupoids, giving more powerful theorems with simpler proofs, and placing the fundamental group into a subsidiary role. So it led in 1967 to the `obvious' question: are groupoids useful in higher homotopy theory? Are there higher homotopy groupoids, and if so, are they useful? Some answers to these questions are referenced on www.bangor.ac.uk/~mas010/hdaweb2.htm , and in the first article in the electronic journal HHA. The van Kampen theorem idea was central to these investigations, as leading to higher dimensional nonabelian local-to-global methods and explicit calculations, some needing computers. So it all started by my trying to understand and give a clear and aesthetic exposition of the van Kampen theorem, suitably generalised to derive the fundamental group of the circle. Does prettier always imply intrinsically more powerful? Thanks to Philip Higgins and other colleagues, and many research students! (Philip Higgins latest (and probably last!) paper is accepted for TAC, and is available as a Bangor preprint.) Here is a project: apply these results on the fundamental groupoid (on a set of base points!!) to braid groups and configuration spaces. Would it help? A search on the arXiv gives other papers on the van Kampen theorem. Ronnie Brown www.bangor.ac.uk/~mas010 www.popmath.org.uk ______________________________________________________________ Subject: van Kampen theprem 2 From: "Ronald Brown" Date: Tue, 8 Feb 2005 23:12:11 -0000 Peter May writes: >> The proof is >> cleaner than the usual proof of the standard connected >> form of van Kampen, since one need not choose all those >> silly paths connecting vertices to the basepoint. The >> standard form ``follows formally''. A good illustration of >> categorical ideas in action in elementary algebraic topology. We found that for a theorem for the fundamental groupoid on a set A of base points and an *arbitrary* open cover, and the right connectivity conditions, one needs to join points of a subdivided path to points of A: see (with A. RAZAK), ``A van Kampen theorem for unions of non-connected spaces'', {\em Archiv. Math.} 42 (1984) 85-88. (The set A has to meet each path component of each 3-fold intersection of elements of the open cover.) Also, it is this form of proof which generalises to higher dimensions, given suitable cubical higher homotopy groupoids. Ronnie Brown __________________________________________________________________ Subject: Re: four responses From: Tim Porter Date: Wed, 09 Feb 2005 09:40:38 +0000 Dear All, My response time this time is less good as there was a night in between! Phil Higgins book mentioned by Philippe Gaucher is available at http://www.tac.mta.ca/tac/reprints/ in electronic form. It has a lot of material in it that is very relevant to problems still around today and deserves to be used more often. That site also has other useful reprints which people may find useful. Tim Don Davis wrote: > Four quick response to this morning's question. (The response by > Porter was received within about 5 minutes of posting.)....DMD > Subject: Re: four postings > From: Philippe Gaucher > Date: Tue, 8 Feb 2005 14:03:21 +0100 > > Maybe (I dont have the book with me) the following book could be helpful for > you : > > @book {MR48:6288, > AUTHOR = {Higgins, P. J.}, > TITLE = {Notes on categories and groupoids}, > NOTE = {Van Nostrand Rienhold Mathematical Studies, No. 32}, > PUBLISHER = {Van Nostrand Reinhold Co.}, > ADDRESS = {London}, > YEAR = {1971}, > PAGES = {v+178}, > MRCLASS = {20L05 (18B10)}, > MRNUMBER = {48 \#6288}, > MRREVIEWER = {V. A. Artamonov}, > } __________________________________________________________________ Subject: Re: four responses From: Yuli Rudyak Date: Tue, 8 Feb 2005 13:21:48 -0500 (EST) Please, post my thanks to all who answered my question. Yuli Dr. Yuli B. Rudyak Department of Mathematics University of Florida