From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7369 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.algtop,gmane.science.mathematics.categories Subject: discrete vector fields (R. Forman) and markings on cell complexes (D.W. Jones) Date: Fri, 06 Jul 2012 07:52:50 +0100 Message-ID: <4FF68B42.4020206@btinternet.com> NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="===============8469413754357472837==" X-Trace: dough.gmane.org 1341574042 8516 80.91.229.3 (6 Jul 2012 11:27:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 6 Jul 2012 11:27:22 +0000 (UTC) To: algtop , category bulletin Original-X-From: algtop-l-bounces+gsma-algtop-l=m.gmane.org-wE+tr93vHrabo6XCN/16Dg@public.gmane.org Fri Jul 06 13:27:20 2012 Return-path: Envelope-to: gsma-algtop-l@m.gmane.org Original-Received: from astroiii.cc.lehigh.edu ([128.180.39.23]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Sn6gx-0005H7-2x for gsma-algtop-l@m.gmane.org; Fri, 06 Jul 2012 13:27:19 +0200 Original-Received: from astroiii.CC.Lehigh.EDU (localhost [127.0.0.1]) by astroiii.cc.lehigh.edu (Postfix) with ESMTP id C0ED4407E9 for ; Fri, 6 Jul 2012 07:27:07 -0400 (EDT) Original-Received: from nm6.bullet.mail.ird.yahoo.com (nm6.bullet.mail.ird.yahoo.com [77.238.189.63]) by astroiii.cc.lehigh.edu (Postfix) with SMTP id B6D9E40016 for ; Fri, 6 Jul 2012 02:52:53 -0400 (EDT) Original-Received: from [77.238.189.230] by nm6.bullet.mail.ird.yahoo.com with NNFMP; 06 Jul 2012 06:52:52 -0000 Original-Received: from [212.82.108.224] by tm11.bullet.mail.ird.yahoo.com with NNFMP; 06 Jul 2012 06:52:52 -0000 Original-Received: from [127.0.0.1] by omp1001.bt.mail.ird.yahoo.com with NNFMP; 06 Jul 2012 06:52:52 -0000 X-Yahoo-Newman-Id: 864523.74773.bm-CZKVv+5DniBNUYSC8IruhW/Tuy6vdkpuoEqcbCQ9QuM@public.gmane.org Original-Received: (qmail 87524 invoked from network); 6 Jul 2012 06:52:52 -0000 DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=s1024; d=btinternet.com; h=DKIM-Signature:X-Yahoo-Newman-Property:X-YMail-OSG:X-Yahoo-SMTP:Received:Message-ID:Date:From:User-Agent:MIME-Version:To:Subject:Content-Type; b=TRg/tMsePne4zkFsEE5ZQOoS/6z9Ft/DgSft+66Ow0KweyHxPalyZy98b08DOAJ+pgYEedaNH5JgkbX0DOoQzrru0hm2tlfgBXmqFh4TEb9a3Q7PpvrEF+FHdw3SQjxhhTtZ63Jf4UWOLOGdYUG0mHGs+GRKPJDQVRk+d8B530A= ; DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=btinternet.com; s=s1024; t=1341557572; bh=feOmrC0W/c9xbewAeC+voeUEkyb/8Q15B9OSbnVHePg=; h=X-Yahoo-Newman-Property:X-YMail-OSG:X-Yahoo-SMTP:Received:Message-ID:Date:From:User-Agent:MIME-Version:To:Subject:Content-Type; b=Wqw33ZxxPsk7ugp7vaQ8OJV2/HhsBrDeehlxolarYkuy1W1M5t6twURW7Y3vw0StMmmL/7eS/lIn6YmJq5R1qAf44xX7N/gq4h73WLJGVKqbgD/O86y5jqEvI7gZpnLbjBJfjyMZnYr4ir5TABA0O7/LJGXZWj0PqvgjsDFE9Pc= X-Yahoo-Newman-Property: ymail-3 X-YMail-OSG: dA1FYXUVM1k5FeejBRHyJOo7Oj.PLEWbywZ__bzk9AGDjUh tFRUY1_oWKbga6NpG4x.U7pkdmSaOBwOzNAogE5ZjiXlM5DK287JaXmu1vbj vhMmXNuK5O3bUnZ5ryUFQw055rYRSEIYqLwrLNqPeX0ewTa_bTWBPxZ8so_S XzSN930DsZbpw6ehJQScxZ3BoEIs_qpZIpaPVVFs9T3mZpycLcdsgAYBiSnt vBiJmvLyryYA3I4ZOXw0yG9cPMGo6VO3wI8gGVKPfL.sA2WbHCAYqPttgMqF TjFQg7eG4zcCdSnSTC5Dx3RwdDmo7LZFUghVErz33DU_fmTQr0rzj1apH0KF hENb9.Gm3.kE44F5iSnM2vbw0GelUgfYPRqd0ot54iqPMb22HWKFEhXaC5OG wIVcXA_g_yF5VZbpZ76_BNnOHJNK67nsTBvN11gY3Hda7f1mKufMatdjCEwh 6lT2Qo6PNIJPDm5MF29T2Uka7BAmCKYaf.QG_AdcJwAAPbMNAzFWW2BvrdUF awPql2u27bmyCAxaG88VIJpYM3DqyE6q4rLQ1sVaq0flWR61OXgSniu_ao92 EZDzVdGW54UJhQC5F X-Yahoo-SMTP: Wmwi34mswBBHdDHVHca64q6T_ef9aH.nrBHTBfugDAmn__Tuj5h6brgWZt0- Original-Received: from [192.168.1.65] (ronnie.profbrown-soZNDvfjGgQIRby/8wo/gg@public.gmane.org with plain) by smtp821.mail.ird.yahoo.com with SMTP; 06 Jul 2012 06:52:52 +0000 UTC User-Agent: Mozilla/5.0 (Windows NT 6.1; WOW64; rv:13.0) Gecko/20120614 Thunderbird/13.0.1 X-Mailman-Approved-At: Fri, 06 Jul 2012 07:26:59 -0400 X-BeenThere: algtop-l-wE+tr93vHrabo6XCN/16Dg@public.gmane.org X-Mailman-Version: 2.1.14 Precedence: list List-Id: Algebraic Topology Discussion Group List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Errors-To: algtop-l-bounces+gsma-algtop-l=m.gmane.org-wE+tr93vHrabo6XCN/16Dg@public.gmane.org Original-Sender: algtop-l-bounces+gsma-algtop-l=m.gmane.org-wE+tr93vHrabo6XCN/16Dg@public.gmane.org Xref: news.gmane.org gmane.science.mathematics.algtop:416 gmane.science.mathematics.categories:7369 Archived-At: This is a multi-part message in MIME format. --===============8469413754357472837== Content-Type: multipart/alternative; boundary="------------030509020003020505070509" This is a multi-part message in MIME format. --------------030509020003020505070509 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Graham Ellis has drawn my attention to the notion of *discrete vector field* on cell complexes due to R. Forman, and on which a number of papers may be found. Graham has some notes for the lecture he gave at Bedlewo, June 24-30, "Dynamics, Topology, Computation". http://hamilton.nuigalway.ie/Berlin/bedlewo.pdf which uses this notion. I found, and Graham agreed, that this notion is equivalent to the notion of "marking" for a cell complex developed in David W Jones Bangor PhD Thesis, (1984), published as "A general theory of polyhedral sets and the corresponding {$T$}-complexes}.{Dissertationes Math. (Rozprawy Mat.)} \textbf{266} (1988) 110pp.*MR0968920* The initial problem solved in this thesis was to define an appropriate notion of "polycell" which would allow a notion of "poly-set" as a contravariant functor from a category of poly cells and inclusions of faces, these polycells to be sufficiently general to include, say, rhombic dodecahedra, and also the cells which occur in van Kampen diagrams for groups. (One thinks also of Stasheff's polytopes.) Such cells had to be regular cell complexes, with one top dimensional cell, and to rigidify the notion such a cell, and all it's subcells, were to be given the structure of a cone on its boundary. Then, and this is the key notion, each cell was to have a marked (distinguished) face. This is equivalent to an arrow pointing from the interior of the cell to that marked face (the opposite direction to Forman's notion!). A final condition required for the further development of the theory was that of shellability. There is more discussion of this work on http://ncatlab.org/nlab/show/T-complex Thus the marking notion is seen to be equivalent to Forman's notion. I feel the reaction should be that if a notion occurs independently and from quite different viewpoints, then the notion is not just twice as good but maybe four or more times as good! (I got this kind of gut feeling in 1967 when George Mackey told me of his work on groupoids in ergodic theory, after a lecture mine on the fundamental groupoid and van Kampen's theorem.) I hope people will be able to look at David Jones' thesis, and other related work he developed there, and see if the work is useful in the increasing applications of discrete vector fields. I hope also the motivation behind his thesis can be developed further. Are these methods related to those of opetopes in higher category theory? Ronnie Brown --------------030509020003020505070509 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Graham Ellis has drawn my attention to the notion of discrete vector field on cell complexes due to R. Forman, and on which a number of papers may be found. Graham has some notes for the lecture he gave at Bedlewo, June 24-30, "Dynamics, Topology, Computation". 
 http://hamilton.nuigalway.ie/Berlin/bedlewo.pdf

which uses this notion. 

I found, and Graham agreed,  that this notion is equivalent to the notion of "marking" for a cell complex developed in David W Jones Bangor PhD Thesis, (1984), published as 

"A general theory of polyhedral sets and the corresponding {$T$}-complexes}.{Dissertationes Math. (Rozprawy Mat.)} \textbf{266} (1988) 110pp. MR0968920

The initial problem solved in this thesis was to define an appropriate notion of "polycell" which would allow a notion of "poly-set" as a contravariant functor from a category of poly cells and inclusions of faces, these polycells to be sufficiently general to include,  say, rhombic dodecahedra, and also the cells which occur in van Kampen diagrams for groups. (One thinks also of Stasheff's polytopes.) 

Such cells had to be regular cell complexes, with one top dimensional cell, and to rigidify the notion such a cell, and all it's subcells,  were to be given the structure of a cone on its boundary. 

Then, and this is the key notion, each cell was to have a marked (distinguished) face. This is equivalent to an arrow pointing from the interior of the cell to that marked face (the opposite direction to Forman's notion!). A final condition required for the further development of the theory was that of shellability. There is more discussion of this work on http://ncatlab.org/nlab/show/T-complex

Thus the marking notion is seen to be equivalent to Forman's notion. I feel the reaction should be that if a notion occurs independently and from quite different viewpoints, then the notion is not just twice as good but maybe four or more times as good! (I got this kind of gut feeling in 1967 when George Mackey told me of his work on groupoids in ergodic theory, after a lecture mine on the fundamental groupoid and van Kampen's theorem.) 

I hope people will be able to look at David Jones' thesis, and other related work he developed there,  and see if the work is  useful in the increasing applications of discrete vector fields. I hope  also the motivation behind his thesis can be developed further. 

Are these methods related to those of opetopes in higher category theory? 

Ronnie Brown


--------------030509020003020505070509-- --===============8469413754357472837== Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline _______________________________________________ ALGTOP-L mailing list ALGTOP-L-wE+tr93vHrabo6XCN/16Dg@public.gmane.org https://lists.lehigh.edu/mailman/listinfo/algtop-l --===============8469413754357472837==--