Two more comments on Mark Hovey's problem list.........DMD Date: Fri, 5 Mar 1999 18:37:42 +0000 (GMT) From: Ronnie Brown Subject: Re: problems I was educated (as much as was possible for me) by someone who defined basic concepts of stable homotopy theory, simple homotopy types (i.e. the beginnings of algebraic K-theory), n-types (including 2-types and crossed modules), etc etc. JHCW was interested in various approximations to homotopy theory, and finer notions of homotopy equivalences, based on elementary operations, i.e. including the algorithmic questions. These problems included and sometimes even necessitated the fundamental group. So I cannot adjust to a view that the main focus, even the Main Stream, is or should be what was considered originally one aspect of the problem and one method. I am happy to work in areas where the fundamental group has an interesting role, and in trying to make these aspects of homotopy theory look more like group theory. So I shall continue to look at homotopy theory as, for me, aiming at a non abelian version of homology, and in which calculations can be carried out in a non abelian spirit (non commutative in dimension 1, even more non commutative in higher dimensions). This is expected in the (very?) long run to be related to basic geometric questions in say differential geometry. It is certainly relevant to say TQFTs. I published a list of problems as ``Some problems in non-Abelian homotopical and homological algebra'', {\em Homotopy theory and related topics, Proceedings Kinosaki, 1988}, ed. M. Mimura, Springer Lecture Notes in Math., 1418 (1990) 105-129. some of which interact with other areas, e.g. foliations. The techniques are maybe orthogonal to those relevant to Mark's list of problems. But now we have for example an equivariant theory involving not just homotopy classes but even function spaces of maps to a classifying space of an equivariant crossed complex, so including 2-types and non local coefficients. Crossed complexes have some advantages over chain complexes, if (or IF) you have an interest in non simply connected spaces. It does seem that the term `higher dimensional algebra' I introduced has in itself been influential (e.g. in theoretical physics). So there are areas which are in the traditions of algebraic topology and which are inherently not tractable by stable methods. Also for me a major interest is that the basic problems of homotopy types reveal the need for and construction of new algebraic structures, and that these new structures have already been of proven interest in other areas. To go further back, the Hopf formula for H_2(G) was one of the origins of homological algebra, which itself plays a not inconsiderable role in the work of of A.Grothendieck which led to a solution of the Weil conjectures. A letter from AG says that he always wanted his students to do non abelian cohomology, so finally he got mad and decided to do it himself. But even then ....! Ronnie Prof R. Brown, School of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382475 fax: +44 1248 383663 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ New article: Higher dimensional group theory Symbolic Sculpture and Mathematics: http://www.bangor.ac.uk/SculMath/ Mathematics and Knots: http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm _______________________________________________ Date: Fri, 5 Mar 1999 11:54:42 -0500 (EST) From: James Stasheff Subject: Re: 2 more on problem list there is a whole branch of `cohomological physics' which is indeed very close to rational homotopy theory but also to claissical algebraic deformation theory and with sevral new twists - due to physicists esp. Batalin, Fradkin and Vilkovisky they combine Chevalley- Eilenberg with koszul-Tate a key difference is that instead of starting over Q one starts over a smooth alg .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds