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: Prof T Porter Date: Tue, 08 Feb 2005 11:56:34 +0000 Dear Yuli, Try Ronnie Brown's /Topology/. The discussion occurs in Chapter 8. Alternatively look up his paper `Groupoids and Van Kampen's theorem' Proc. London Math. Soc (3) 17 (1967) 385-340. Tim Porter > > ______________________________________________________________ > > Subject: question > From: Yuli Rudyak > Date: Mon, 7 Feb 2005 19:04:00 -0500 (EST) > > Where can I find the van Kampen Theorem for the case when the intersection of > parts is not connected (like as we divide a circle in two segments and > intersection has two components). > > Yuli > > Dr. Yuli B. Rudyak > Department of Mathematics > University of Florida > 358 Little Hall _____________________________________________________________________ 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}, } The Van Kampen theorem which is stated in this book is that the fundamental groupoid functor preserves pushouts. The fundamental groupoid functor consists of all continuous paths up to homotopy. Sincerely yours. pg. __________________________________________________________ Subject: topology discussion list From: Fernando Muro Date: Tue, 8 Feb 2005 16:02:16 +0100 (MET) Brown, Ronald Topology. A geometric account of general topology, homotopy types and the fundamental groupoid. Second edition. Ellis Horwood Series: Mathematics and its Applications. Best regards. Fernando Muro. ___________________________________________________________ Subject: Van kampen From: Peter May Date: Tue, 8 Feb 2005 09:30:35 -0600 Well, if Allen can shamelessly advertise his book ... The van Kampen theorem without a condition that the relevant intersections be connected is properly a statement about fundamental groupoids rather than fundamental groups. In the statement of the fundamental groupoid version of van Kampen on page 17 of ``A concise course in algebraic topology'', I start with an open cover of path connected subsets closed under intersection, but one can drop the words ``path connected'' with no change whatsoever in the statement or the proof. 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. Peter May